Raster multiplexing in photonic circuits

ABSTRACT

Circuits and methods that implement multiplexing for photons propagating in waveguides are disclosed, in which an input photon received on a selected one of a set of input waveguides can be selectably routed to one of a set of output waveguides. The output waveguide can be selected on a rotating or cyclic basis, in a fixed order, and the input waveguide can be selected based at least in part on which one(s) of a set of input waveguides is (are) currently propagating a photon.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.17/305,024, filed Jun. 29, 2021, which claims priority to U.S.Application No. 63/047,093, filed Jul. 1, 2020, and to U.S. ApplicationNo. 63/047,731, filed Jul. 2, 2020, the disclosures of which areincorporated herein by reference.

BACKGROUND

In photonic circuits and systems, photons may be generated at differenttimes and propagated through different waveguides. For variousoperations, it may be desirable to rearrange photons spatially ontodifferent waveguides and/or to synchronize photons propagating ondifferent waveguides so that they arrive concurrently at a particularlocation within the circuit.

SUMMARY

Disclosed herein are examples (also referred to as “embodiments”) ofcircuits and methods that implement multiplexing in photonic circuits.An input photon received on a selected one of a set of input waveguidescan be selectably routed to one of a set of output waveguides. Theoutput waveguide can be selected on a rotating or cyclic basis, in afixed order, and the input waveguide can be selected based at least inpart on which one(s) of a set of input waveguides is (are) currentlypropagating a photon. In some embodiments, there may be just one inputwaveguide that is always selected.

Some embodiments relate to a circuit that can comprise a number (N) ofinput paths of input paths, where N is at least 1; a number of outputpaths including a raster group of output paths, where the raster groupof output paths has a number (R) of output paths, where R is at least 2;an optical switching network coupled between the input paths and theoutput paths, the optical switching network comprising a plurality ofactive optical switches arranged to selectably couple a photon from anyone of the input paths to any one of the output paths; and control logiccoupled to the optical switching network. The control logic can beconfigured to: receive an input signal indicative of when a photon ispresent on each input path; select one of the output paths as an activeoutput path, wherein output paths in the raster group are selectedaccording to a fixed order; and generate control signals to set a stateof the active optical switches such that a photon from one of the inputpaths is coupled to the active output path.

In some embodiments, each output path in the raster group of outputpaths is selected as the active output path once during a raster periodconsisting of R consecutive time bins.

In some embodiments, the number N of input paths is greater than 1 andthe control logic is further configured to select one of the input pathsas an active input path based on the input signal and to generate thecontrol signals such that a photon from the active input path is coupledto the active output path.

In some embodiments, the circuit can also include a number of delaylines, each delay line introducing a different amount of delay, and eachdelay line can be coupled to a different one of the R output paths inthe raster group of output paths. The control logic can be configured toselect the output paths in an order such that photons entering theoptical switching network during a set of R consecutive time bins arriveat respective outputs of the delay lines in the same time bin.

In some embodiments, the optical switching network can be a generalizedMach-Zehnder interferometer (GMZI), and the active optical switches caninclude active phase shifters.

In some embodiments, each input path and each output path comprises awaveguide. In other embodiments, each input path and each output pathcomprises a pair of waveguides. In still other embodiments, each inputpath and each output path comprises a number of waveguides that islarger than two.

In some embodiments, each input path can be coupled to an output of adifferent one of a set of N heralded single photon sources, and theinput signal can include heralding signals from the heralded singlephoton sources.

In some embodiments, the output paths further include at least oneadditional output path separate from the raster group of output paths.

Some embodiments relate to a circuit that comprises a number (N) ofsource circuits, each source circuit having an output path to propagatea photon, where N is at least 1; a downstream circuit having a number(R) of input paths to receive photons, where R is at least 2; and araster multiplexer circuit. The raster multiplexer circuit can include:a number N of multiplexer input paths, each multiplexer input pathcoupled to an output path of one of the source circuits; a number ofmultiplexer output paths including a raster group of multiplexer outputpaths, wherein the raster group of multiplexer output paths includes Rmultiplexer output paths, each multiplexer output path in the rastergroup of multiplexer output paths being coupled to one of the inputpaths of the downstream circuit; an optical switching network coupledbetween the multiplexer input paths and the multiplexer output paths,the optical switching network comprising a set of active opticalswitches arranged to selectably couple a photon from any one of themultiplexer input paths to any one of the multiplexer output paths; andcontrol logic coupled to the optical switching network. The controllogic can be configured to: receive an input signal indicative of whenthe output path of each source circuit is propagating a photon; selectone of the multiplexer output paths as an active multiplexer outputpath, where each multiplexer output path in the raster group ofmultiplexer output paths is selected as the active raster multiplexeroutput path once during a raster period consisting of R consecutive timebins; and generate control signals to set a state of the optical activeswitches such that a photon from one of the multiplexer input paths iscoupled to the active multiplexer output path.

In some embodiments, the number N can be greater than 1 and the controllogic can be further configured to: select one of the multiplexer inputpaths as an active multiplexer input path based on the input signal; andgenerate the control signals such that a photon from the activemultiplexer input path is coupled to the active multiplexer output path.

In some embodiments, the downstream circuit can be a Bell stategenerator.

In some embodiments, the source circuits can be heralded single photonsource circuits. In other embodiments, the source circuits can beentanglement circuits that generate entangled systems of photons thatencode qubits. For example, the qubits can be encoded using a dual-railencoding, and each multiplexer input path and each multiplexer outputpath can include a pair of waveguides.

In some embodiments, the downstream circuit can include a second opticalswitching network coupled to a plurality of fusion circuits.

Some embodiments relate to a circuit that includes: a number (N) ofsource circuits, each source circuit having an output path to propagatea photon; a number (R) of downstream circuits, each downstream circuithaving a number (V) of input paths to receive photons, where R is atleast 2 and M is at least 2; a set of M raster multiplexer circuits; andcontrol logic coupled to the raster multiplexer circuits. Each rastermultiplexer circuit can include: a set of N multiplexer input paths,each multiplexer input path coupled to an output path of one of the Nsource circuits; a number of multiplexer output paths including a rastergroup of multiplexer output paths, wherein the raster group ofmultiplexer output paths includes R multiplexer output paths, eachraster multiplexer output path in the raster group of multiplexer outputpaths being coupled to one of the input paths of a different one of theR downstream circuits; and an optical switching network coupled betweenthe multiplexer input paths and the multiplexer output paths, theoptical switching network comprising a plurality of active opticalswitches arranged to selectably couple a photon from any one of themultiplexer input paths to any one of the multiplexer output paths. Thecontrol logic can be configured to: receive an input signal indicativeof when the output path of each source circuit is propagating a photon;select, for each of the raster multiplexer circuits, one of themultiplexer input paths as an active multiplexer input path, theselection being based at least in part on the input signal; select, foreach of the raster multiplexer circuits, one of the multiplexer outputpaths as an active multiplexer output path such that, for each rastermultiplexer circuit, each multiplexer output path in the raster group ofmultiplexer output paths is selected as the active multiplexer outputpath once during a raster period consisting of R consecutive time binsand such that all M of the multiplexer output paths that couple to asame one of the R downstream circuits are selected as the activemultiplexer output paths for a same time bin; and generate controlsignals to set a state of the active switches in the optical switchingnetwork of each of the R raster multiplexer circuits such that, in eachof the R raster multiplexer circuits, a photon from the activemultiplexer input path is coupled to the active multiplexer output path.

In some embodiments, the circuit can also include a set of delay lines,each delay line introducing a different amount of delay, and each delayline can be coupled to a different one of the multiplexer output pathsin the raster group of multiplexer output paths. The control logic canbe further configured to select the output paths in an order such thatphotons entering the optical switching network during a set of Rconsecutive time bins arrive at respective outputs of the delay lines inthe same time bin.

In some embodiments, the source circuits can be heralded single photonsource circuits, and each downstream circuit can be a Bell stategenerator.

Some embodiments relate to a method that can include: receiving a set ofinput signals indicating whether photons are present on each of a set ofinput paths of an optical circuit; selecting an active input path forthe optical circuit based at least in part on the input signals;selecting an active output path for the optical circuit from a number ofoutput paths that includes a raster group of a number (R) of outputpaths, wherein R is at least 2, wherein output paths in the raster groupare selected according to a fixed order; and controlling a set of activeswitches in the optical circuit to couple a photon from the active inputpath to the active output path.

In some embodiments, each output path in the raster group can beselected as the active output path once during a raster periodconsisting of R consecutive clock cycles. In some embodiments, eachoutput path in the raster group of output paths can be coupled to adelay circuit that introduces a different number of clock cycles ofdelay, and the output paths can be selected in an order such thatphotons entering the optical circuit during a set of R consecutivecycles arrive at respective outputs of the delay lines in a same clockcycle.

The following detailed description, together with the accompanyingdrawings, will provide a better understanding of the nature andadvantages of the claimed invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows two representations of a portion of a pair of waveguidescorresponding to a dual-rail-encoded photonic qubit.

FIG. 2A shows a schematic diagram for coupling of two modes.

FIG. 2B shows, in schematic form, a physical implementation of modecoupling in a photonic system that can be used in some embodiments.

FIGS. 3A and 3B show, in schematic form, examples of physicalimplementations of a Mach-Zehnder Interferometer (MZI) configurationthat can be used in some embodiments.

FIG. 4A shows another schematic diagram for coupling of two modes.

FIG. 4B shows, in schematic form, a physical implementation of the modecoupling of FIG. 4A in a photonic system that can be used in someembodiments.

FIG. 5 shows a four-mode coupling scheme that implements a “spreader,”or “mode-information erasure,” transformation on four modes inaccordance with some embodiments.

FIG. 6 illustrates an example optical device that can implement thefour-mode mode-spreading transform shown schematically in FIG. 5 inaccordance with some embodiments.

FIG. 7 shows a circuit diagram for a dual-rail-encoded Bell stategenerator that can be used in some embodiments.

FIG. 8A shows a circuit diagram for a dual-rail-encoded type I fusiongate that can be used in some embodiments.

FIG. 8B shows example results of type I fusion operations using the gateof FIG. 8A.

FIG. 9A shows a circuit diagram for a dual-rail-encoded type II fusiongate that can be used in some embodiments.

FIG. 9B shows an example result of a type II fusion operation using thegate of FIG. 9A.

FIG. 10 illustrates an example of a qubit entangling system inaccordance with some embodiments.

FIG. 11 shows an example of an N×1 spatial multiplexing circuit for aset of N photon sources.

FIG. 12 shows a simplified schematic view of a raster multiplexingcircuit according to some embodiments.

FIG. 13 shows a flow diagram of a process according to some embodiments.

FIG. 14 shows a simplified schematic view of an optical circuit thatincludes a raster multiplexing circuit coupled to a Bell state generatoraccording to some embodiments.

FIG. 15 shows a simplified schematic view of an optical circuit thatincludes two raster multiplexing circuits coupled to a Bell stategenerator according to some embodiments.

FIGS. 16A-16C show how a raster multiplexing circuit can be used toenable a single copy of an “upstream” circuit used to provide inputs toa “downstream” circuit according to some embodiments.

FIG. 17 shows a simplified schematic diagram of an optical circuitaccording to some embodiments.

FIG. 18 shows a simplified schematic view of an optical circuitaccording to some embodiments.

FIGS. 19A and 19B together show a simplified circuit schematic of anoptical circuit according to some embodiments.

FIG. 20 is a spacetime diagram further illustrating the operation of thecircuit of FIGS. 19A and 19B according to some embodiments.

FIGS. 21A and 21B show building blocks of composite switch networks thatcan be used in some embodiments.

FIG. 21C shows a N-to-M GMZI that can be used in some embodiments.

FIGS. 22A and 22B show spatial N-to-1 muxes, with inputs at Nspatially-distinct locations (ports), that can be used in someembodiments.

FIGS. 23A and 23B show N-to-1 temporal muxes, with inputs in N distincttime bins, that can be used in some embodiments.

FIGS. 24A-24D show examples of generalized N-to-1 composite multiplexingnetworks that can be used in some embodiments.

FIGS. 25A and 25B show examples of N-to-M switch networks that can beused in some embodiments.

FIG. 26 shows an equation for a type of specific decomposition of GMZInetworks that can be used in some embodiments.

FIGS. 27A and 27B show Hadamard-type GMZI constructions that can be usedin some embodiments.

FIGS. 28A and 28B show examples of larger GMZI that can be used in someembodiments.

DETAILED DESCRIPTION

Disclosed herein are examples (also referred to as “embodiments”) ofcircuits and methods that implement multiplexing for photons propagatingin waveguides. An input photon received on a selected one of a set ofinput waveguides can be selectably routed to one of a set of outputwaveguides. The output waveguide can be selected on a rotating or cyclicbasis, in a fixed order, and the input waveguide can be selected basedat least in part on which one(s) of a set of input waveguides is (are)currently propagating a photon. (In some embodiments, there may be justone input waveguide that is always selected.)

Circuits and methods of the kind described herein can be used in avariety of applications where spatial multiplexing is desired. Tofacilitate understanding of the disclosure, an overview of relevantconcepts and terminology is provided in Section 1. Section 2 introducesspatial multiplexing techniques for photons in waveguides. Sections 3and 4 describe “raster” multiplexing techniques according to variousembodiments. Although embodiments are described with specific detail tofacilitate understanding, those skilled in the art with access to thisdisclosure will appreciate that the claimed invention can be practicedwithout these details.

1. Overview of Quantum Computing

Quantum computing relies on the dynamics of quantum objects, e.g.,photons, electrons, atoms, ions, molecules, nanostructures, and thelike, which follow the rules of quantum theory. In quantum theory, thequantum state of a quantum object is described by a set of physicalproperties, the complete set of which is referred to as a mode. In someembodiments, a mode is defined by specifying the value (or distributionof values) of one or more properties of the quantum object. For example,in the case where the quantum object is a photon, modes can be definedby the frequency of the photon, the position in space of the photon(e.g., which waveguide or superposition of waveguides the photon ispropagating within), the associated direction of propagation (e.g., thek-vector for a photon in free space), the polarization state of thephoton (e.g., the direction (horizontal or vertical) of the photon'selectric and/or magnetic fields), a time window in which the photon ispropagating, the orbital angular momentum state of the photon, and thelike.

For the case of photons propagating in a waveguide, it is convenient toexpress the state of the photon as one of a set of discretespatio-temporal modes. For example, the spatial mode k of the photon isdetermined according to which one of a finite set of discrete waveguidesthe photon is propagating in, and the temporal mode t_(j) is determinedby which one of a set of discrete time periods (referred to herein as“bins”) the photon is present in. In some photonic implementations, thedegree of temporal discretization can be provided by a pulsed laserwhich is responsible for generating the photons. As used herein, termssuch as “simultaneous” or “concurrent” refer to events occurring withinthe same time bin, and terms such as “synchronous” (or “synchronized”)refer to events separated by a predictable, constant number of timebins, which can but need not be zero. The term “path” is used herein torefer to a set of one or more waveguides representing spatial modes, anddepending on how the photons are being used, a path may include one ormore waveguides. In examples below, spatial modes will be used primarilyto avoid complication of the description. However, one of ordinary skillwill appreciate that the systems and methods can apply to any type ofmode, e.g., temporal modes, polarization modes, and any other mode orset of modes that serves to specify the quantum state. Further, in thedescription that follows, embodiments will be described that employphotonic waveguides to define the spatial modes of the photon. However,persons of ordinary skill in the art with access to this disclosure willappreciate that other types of mode, e.g., temporal modes, energystates, and the like, can be used without departing from the scope ofthe present disclosure. In addition, persons of ordinary skill in theart will be able to implement examples using other types of quantumsystems, including but not limited to other types of photonic systems.

For quantum systems of multiple indistinguishable particles, rather thandescribing the quantum state of each particle in the system, it isuseful to describe the quantum state of the entire many-body systemusing the formalism of Fock states (sometimes referred to as theoccupation number representation). In the Fock state description, themany-body quantum state is specified by how many particles there are ineach mode of the system. For example, a multimode, two particle Fockstate |1001

_(1,2,3,4) specifies a two-particle quantum state with one particle inmode 1, zero particles in mode 2, zero particles in mode 3, and oneparticle in mode 4. Again, as introduced above, a mode can be anyproperty of the quantum object. For the case of a photon, any two modesof the electromagnetic field can be used, e.g., one may design thesystem to use modes that are related to a degree of freedom that can bemanipulated passively with linear optics. For example, polarization,spatial degree of freedom, or angular momentum could be used. Thefour-mode system represented by the two particle Fock state |1001

_(1,2,3,4) can be physically implemented as four distinct waveguideswith two of the four waveguides having one photon travelling withinthem. Other examples of a state of such a many-body quantum systeminclude the four-particle Fock state |1111

_(1,2,3,4) that represents each mode occupied by one particle and thefour-particle Fock state |2200

_(1,2,3,4) that represents modes 1 and 2 respectively occupied by twoparticles and modes 3 and 4 occupied by zero particles. For modes havingzero particles present, the term “vacuum mode” is used. For example, forthe four-particle Fock state |2200

_(1,2,3,4) modes 3 and 4 are referred to herein as “vacuum modes.” Fockstates having a single occupied mode can be represented in shorthandusing a subscript to identify the occupied mode. For example, |0010

_(1,2,3,4) is equivalent to |1₃

.

1.1.Qubits

As used herein, a “qubit” (or quantum bit) is a quantum system with anassociated quantum state that can be used to encode information. Aquantum state can be used to encode one bit of information if thequantum state space can be modeled as a (complex) two-dimensional vectorspace, with one dimension in the vector space being mapped to logicalvalue 0 and the other to logical value 1. In contrast to classical bits,a qubit can have a state that is a superposition of logical values 0and 1. More generally, a “qudit” can be any quantum system having aquantum state space that can be modeled as a (complex) n-dimensionalvector space (for any integer n), which can be used to encode n bits ofinformation. For the sake of clarity of description, the term “qubit” isused herein, although in some embodiments the system can also employquantum information carriers that encode information in a manner that isnot necessarily associated with a binary bit, such as a qudit. Qubits(or qudits) can be implemented in a variety of quantum systems. Examplesof qubits include: polarization states of photons; presence of photonsin waveguides; or energy states of molecules, atoms, ions, nuclei, orphotons. Other examples include other engineered quantum systems such asflux qubits, phase qubits, or charge qubits (e.g., formed from asuperconducting Josephson junction); topological qubits (e.g., Majoranafermions); or spin qubits formed from vacancy centers (e.g., nitrogenvacancies in diamond).

A qubit can be “dual-rail encoded” such that the logical value of thequbit is encoded by occupation of one of two modes of the quantumsystem. For example, the logical 0 and 1 values can be encoded asfollows:

|0

_(L)=|10

_(1,2)  (1)

|1

_(L)=|01

_(1,2)  (2)

where the subscript “L” indicates that the ket represents a logicalstate (e.g., a qubit value) and, as before, the notation |ij

_(1,2) on the right-hand side of the equations above indicates thatthere are i particles in a first mode and j particles in a second mode,respectively (e.g., where i and j are integers). In this notation, atwo-qubit system having a logical state |0

|1

_(L) (representing a state of two qubits, the first qubit being in a ‘0’logical state and the second qubit being in a ‘1’ logical state) may berepresented using occupancy across four modes by |1001

_(1,2,3,4) (e.g., in a photonic system, one photon in a first waveguide,zero photons in a second waveguide, zero photons in a third waveguide,and one photon in a fourth waveguide). In some instances throughout thisdisclosure, the various subscripts are omitted to avoid unnecessarymathematical clutter.

1.2.Entangled States

Many of the advantages of quantum computing relative to “classical”computing (e.g., conventional digital computers using binary logic) stemfrom the ability to create entangled states of multi-qubit systems. Inmathematical terms, a state |ψ

of n quantum objects is a separable state if |ψ

=|ψ₁

⊗ . . . ⊗|ψ_(n)

, and an entangled state is a state that is not separable. One exampleis a Bell state, which, loosely speaking, is a type of maximallyentangled state for a two-qubit system, and qubits in a Bell state maybe referred to as a Bell pair. For example, for qubits encoded by singlephotons in pairs of modes (a dual-rail encoding), examples of Bellstates include:

$\begin{matrix}{\left. {❘\Phi^{+}} \right\rangle = {\frac{\left. {\left. {\left. {\left. {❘0} \right\rangle_{L}{❘0}} \right\rangle_{L} + {❘1}} \right\rangle_{L}{❘1}} \right\rangle_{L}}{\sqrt{2}} = \frac{\left. {\left. {\left. {\left. {❘10} \right\rangle{❘10}} \right\rangle + {❘01}} \right\rangle{❘01}} \right\rangle}{\sqrt{2}}}} & (3)\end{matrix}$ $\begin{matrix}{\left. {❘\Phi^{-}} \right\rangle = {\frac{\left. {\left. {\left. {\left. {❘0} \right\rangle_{L}{❘0}} \right\rangle_{L} - {❘1}} \right\rangle_{L}{❘1}} \right\rangle_{L}}{\sqrt{2}} = \frac{\left. {\left. {\left. {\left. {❘10} \right\rangle{❘10}} \right\rangle - {❘01}} \right\rangle{❘01}} \right\rangle}{\sqrt{2}}}} & (4)\end{matrix}$ $\begin{matrix}{\left. {❘\Psi^{+}} \right\rangle = {\frac{\left. {\left. {\left. {\left. {❘0} \right\rangle_{L}{❘1}} \right\rangle_{L} + {❘1}} \right\rangle_{L}{❘0}} \right\rangle_{L}}{\sqrt{2}} = \frac{\left. {\left. {\left. {\left. {❘10} \right\rangle{❘01}} \right\rangle + {❘01}} \right\rangle{❘10}} \right\rangle}{\sqrt{2}}}} & (5)\end{matrix}$ $\begin{matrix}{\left. {❘\Psi^{-}} \right\rangle = {\frac{\left. {\left. {\left. {\left. {❘0} \right\rangle_{L}{❘1}} \right\rangle_{L} - {❘1}} \right\rangle_{L}{❘0}} \right\rangle_{L}}{\sqrt{2}} = \frac{\left. {\left. {\left. {\left. {❘10} \right\rangle{❘01}} \right\rangle - {❘01}} \right\rangle{❘10}} \right\rangle}{\sqrt{2}}}} & (6)\end{matrix}$

More generally, an n-qubit Greenberger-Horne-Zeilinger (GHZ) state (or“n-GHZ state”) is an entangled quantum state of n qubits. For a givenorthonormal logical basis, an n-GHZ state is a quantum superposition ofall qubits being in a first basis state superposed with all qubits beingin a second basis state:

$\begin{matrix}{\left. {❘{GHZ}} \right\rangle = \frac{\left. {\left. {❘0} \right\rangle^{\otimes M} + {❘1}} \right\rangle^{\otimes M}}{\sqrt{2}}} & (7)\end{matrix}$

where the kets above refer to the logical basis. For example, for qubitsencoded by single photons in pairs of modes (a dual-rail encoding), a3-GHZ state can be written:

$\begin{matrix}{\left. {❘{GHZ}} \right\rangle = {\frac{\left. {\left. {\left. {\left. {\left. {\left. {❘0} \right\rangle_{L}{❘0}} \right\rangle_{L}{❘0}} \right\rangle_{L} - {❘1}} \right\rangle_{L}{❘1}} \right\rangle_{L}{❘1}} \right\rangle_{L}}{\sqrt{2}} = \frac{\left. {\left. {\left. {\left. {\left. {\left. {❘10} \right\rangle{❘10}} \right\rangle{❘10}} \right\rangle + {❘01}} \right\rangle{❘01}} \right\rangle{❘01}} \right\rangle}{\sqrt{2}}}} & (8)\end{matrix}$

where the kets above refer to photon occupation number in six respectivemodes (with mode subscripts omitted).

1.3.Physical Implementations

Qubits (and operations on qubits) can be implemented using a variety ofphysical systems. In some examples described herein, qubits are providedin an integrated photonic system employing waveguides, beam splitters,photonic switches, and single photon detectors, and the modes that canbe occupied by photons are spatiotemporal modes that correspond topresence of a photon in a waveguide. Modes can be coupled using modecouplers, e.g., optical beam splitters, to implement transformationoperations, and measurement operations can be implemented by couplingsingle-photon detectors to specific waveguides. One of ordinary skill inthe art with access to this disclosure will appreciate that modesdefined by any appropriate set of degrees of freedom, e.g., polarizationmodes, temporal modes, and the like, can be used without departing fromthe scope of the present disclosure. For instance, for modes that onlydiffer in polarization (e.g., horizontal (H) and vertical (V)), a modecoupler can be any optical element that coherently rotates polarization,e.g., a birefringent material such as a waveplate. For other systemssuch as ion trap systems or neutral atom systems, a mode coupler can beany physical mechanism that can couple two modes, e.g., a pulsedelectromagnetic field that is tuned to couple two internal states of theatom/ion.

In some embodiments of a photonic quantum computing system usingdual-rail encoding, a qubit can be implemented using a pair ofwaveguides. FIG. 1 shows two representations (100, 100′) of a portion ofa pair of waveguides 102, 104 that can be used to provide adual-rail-encoded photonic qubit. At 100, a photon 106 is in waveguide102 and no photon is in waveguide 104 (also referred to as a vacuummode); in some embodiments, this corresponds to the |0

_(L) state of a photonic qubit. At 100′, a photon 108 is in waveguide104, and no photon is in waveguide 102; in some embodiments thiscorresponds to the |1

_(L) state of the photonic qubit. To prepare a photonic qubit in a knownlogical state, a photon source (not shown) can be coupled to one end ofone of the waveguides. The photon source can be operated to emit asingle photon into the waveguide to which it is coupled, therebypreparing a photonic qubit in a known state. Photons travel through thewaveguides, and by periodically operating the photon source, a quantumsystem having qubits whose logical states map to different temporalmodes of the photonic system can be created in the same pair ofwaveguides. In addition, by providing multiple pairs of waveguides, aquantum system having qubits whose logical states correspond todifferent spatiotemporal modes can be created. It should be understoodthat the waveguides in such a system need not have any particularspatial relationship to each other. For instance, they can be but neednot be arranged in parallel. In the context of optical circuitsoperating on qubits, a “path” may refer to a set of (one or more)waveguides that provides a set of spatial modes for one qubit. In adual-rail encoding, a path includes a pair of waveguides. Since eachwaveguide in a dual-rail encoding corresponds to a (spatial) mode, theterm “mode” is sometimes used interchangeably with “waveguide” indescriptions of circuits for dual-rail encoded qubits. Other encodingsmay use a different number of waveguides. For instance, a polarizationencoding may use a single waveguide for each path.

Occupied modes can be created by using a photon source to generate aphoton that then propagates in the desired waveguide. A photon sourcecan be, for instance, a resonator-based source that emits photon pairs,also referred to as a heralded single photon source. In one example ofsuch a source, the source is driven by a pump, e.g., a light pulse, thatis coupled into a system of optical resonators that, through a nonlinearoptical process (e.g., spontaneous four wave mixing (SFWM), spontaneousparametric down-conversion (SPDC), second harmonic generation, or thelike), can generate a pair of photons. Many different types of photonsources can be employed. Examples of photon pair sources can include amicroring-based spontaneous four wave mixing (SPFW) heralded photonsource (HPS). However, the precise type of photon source used is notcritical and any type of nonlinear source, employing any process, suchas SPFW, SPDC, or any other process can be used. Other classes ofsources that do not necessarily require a nonlinear material can also beemployed, such as those that employ atomic and/or artificial atomicsystems, e.g., quantum dot sources, color centers in crystals, and thelike. In some cases, sources may or may not be coupled to photoniccavities, e.g., as can be the case for artificial atomic systems such asquantum dots coupled to cavities. Other types of photon sources alsoexist for SFWM and SPDC, such as optomechanical systems and the like.For purposes of the present disclosure, the precise type of photonsource used is not critical and any type of heralded single photonsource, employing any process, such as SPFW, SPDC, or any other process,can be used.

In such cases, operation of the photon source may be non-deterministic(also sometimes referred to as “stochastic”) such that a given pumppulse may or may not produce a photon pair. In some embodiments, when aheralded single photon source generates a pair of photons, one photon ofthe pair can be propagated into a “signaling” (or “propagation”)waveguide of an optical circuit, and the other photon (sometimesreferred to as a “heralding photon”) can be propagated into a differentwaveguide, which can be coupled to a single-photon detector. Thesingle-photon detector can generate a signal (e.g., a digital logicsignal) indicating when a photon has been detected by the detector. Anytype of photodetector that has sensitivity to single photons can beused. In some embodiments, detection of a photon in a particularheralding waveguide indicates presence of a photon in a correspondingsignaling waveguide. Accordingly, it can be known when and where aphoton is generated.

In some embodiments, coherent spatial and/or temporal multiplexing ofseveral non-deterministic sources (referred to herein as “active”multiplexing) can be used to allow the probability of having one modebecome occupied during a given cycle to approach 1. One of ordinaryskill will appreciate that many different active multiplexingarchitectures that incorporate spatial and/or temporal multiplexing arepossible. For instance, active multiplexing schemes that employlog-tree, generalized Mach-Zehnder interferometers, multimodeinterferometers, chained sources, chained sources with dump-the-pumpschemes, asymmetric multi-crystal single photon sources, or any othertype of active multiplexing architecture can be used. In someembodiments, the photon source can employ an active multiplexing schemewith quantum feedback control and the like. In some embodiments, use ofmultirail encoding allows the probability of a band having one modebecome occupied during a given pulse cycle to approach 1 without activemultiplexing. Specific examples of multiplexing operations that can beapplied to non-deterministic photon sources are described below.

Measurement operations can be implemented by coupling a waveguide to asingle-photon detector that generates a classical signal (e.g., adigital logic signal) indicating that a photon has been detected by thedetector. Any type of photodetector that has sensitivity to singlephotons can be used. In some embodiments, detection of a photon (e.g.,at the output end of a waveguide) indicates an occupied mode whileabsence of a detected photon can indicate an unoccupied mode.

Some embodiments described below relate to physical implementations ofunitary transform operations that couple modes of a quantum system,which can be understood as transforming the quantum state of the system.For instance, if the initial state of the quantum system (prior to modecoupling) is one in which one mode is occupied with probability 1 andanother mode is unoccupied with probability 1 (e.g., a state |10

in the Fock notation introduced above), mode coupling can result in astate in which both modes have a nonzero probability of being occupied,e.g., a state a₁|10

+a₂|01

, where |a₁|²+|a₂|²=1. In some embodiments, operations of this kind canbe implemented by using beam splitters to couple modes together andvariable phase shifters to apply phase shifts to one or more modes. Theamplitudes a₁ and a₂ depend on the reflectivity (or transmissivity) ofthe beam splitters and on any phase shifts that are introduced.

FIG. 2A shows a schematic diagram 210 (also referred to as a circuitdiagram or circuit notation) for coupling of two modes. The modes aredrawn as horizontal lines 212, 214, and the mode coupler 216 isindicated by a vertical line that is terminated with nodes (solid dots)to identify the modes being coupled. In the more specific language oflinear quantum optics, the mode coupler 216 shown in FIG. 2A representsa 50/50 beam splitter that implements a transfer matrix:

$\begin{matrix}{{T = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 & i \\i & 1\end{pmatrix}}},} & (9)\end{matrix}$

where T defines the linear map for the photon creation operators on twomodes. (In certain contexts, transfer matrix T can be understood asimplementing a first-order imaginary Hadamard transform.) By conventionthe first column of the transfer matrix corresponds to creationoperators on the top mode (referred to herein as mode 1, labeled ashorizontal line 212), and the second column corresponds to creationoperators on the second mode (referred to herein as mode 2, labeled ashorizontal line 214), and so on if the system includes more than twomodes. More explicitly, the mapping can be written as:

$\begin{matrix}{\left. \begin{pmatrix}a_{1}^{\dagger} \\a_{2}^{\dagger}\end{pmatrix}_{input}\mapsto{\frac{1}{\sqrt{2}}\begin{pmatrix}1 & {- i} \\{- i} & 1\end{pmatrix}\begin{pmatrix}a_{1}^{\dagger} \\a_{2}^{\dagger}\end{pmatrix}_{output}} \right.,} & (10)\end{matrix}$

where subscripts on the creation operators indicate the mode that isoperated on, the subscripts input and output identify the form of thecreation operators before and after the beam splitter, respectively andwhere:

a _(i) |n _(i) ,n _(j)

=√{square root over (n _(i))}|n _(i)−1,n _(j)

a _(j) |n _(j)

=√{square root over (n _(j))}|n _(i) ,n _(j)−1

a _(j) ^(†) |n _(i) ,n _(j)

=√{square root over (n _(j)+1)}|n _(i) ,n _(j)+1)  (11)

For example, the application of the mode coupler shown in FIG. 2A leadsto the following mappings:

$\begin{matrix}\left. a_{1_{input}}^{\dagger}\mapsto{\frac{1}{\sqrt{2}}\left( {a_{1_{output}}^{\dagger} - {ia_{2_{output}}^{\dagger}}} \right)} \right. & (12)\end{matrix}$$\left. a_{2_{input}}^{\dagger}\mapsto{\frac{1}{\sqrt{2}}\left( {{- {ia}_{1_{output}}^{\dagger}} + a_{2_{output}}^{\dagger}} \right)} \right.$

Thus, the action of the mode coupler described by Eq. (9) is to take theinput states |10

, |01

, and |11

to

$\begin{matrix}\left. \left. {❘10} \right\rangle\mapsto\frac{\left. {\left. {❘10} \right\rangle - {i{❘01}}} \right\rangle}{\sqrt{2}} \right. & (13)\end{matrix}$$\left. \left. {❘01} \right\rangle\mapsto\frac{\left. {\left. {{- i}{❘10}} \right\rangle + {❘01}} \right\rangle}{\sqrt{2}} \right.$$\left. \left. {\left. \left. {❘11} \right\rangle\mapsto{\frac{- i}{2}\left( {❘20} \right.} \right\rangle + {❘02}} \right\rangle \right)$

FIG. 2B shows a physical implementation of a mode coupling thatimplements the transfer matrix T of Eq. (9) for two photonic modes inaccordance with some embodiments. In this example, the mode coupling isimplemented using a waveguide beam splitter 200, also sometimes referredto as a directional coupler or mode coupler. Waveguide beam splitter 200can be realized by bringing two waveguides 202, 204 into close enoughproximity that the evanescent field of one waveguide can couple into theother. By adjusting the separation d between waveguides 202, 204 and/orthe length l of the coupling region, different couplings between modescan be obtained. In this manner, a waveguide beam splitter 200 can beconfigured to have a desired transmissivity. For example, the beamsplitter can be engineered to have a transmissivity equal to 0.5 (i.e.,a 50/50 beam splitter for implementing the specific form of the transfermatrix T introduced above). If other transfer matrices are desired, thereflectivity (or the transmissivity) can be engineered to be greaterthan 0.6, greater than 0.7, greater than 0.8, or greater than 0.9without departing from the scope of the present disclosure.

In addition to mode coupling, some unitary transforms may involve phaseshifts applied to one or more modes. In some photonic implementations,variable phase-shifters can be implemented in integrated circuits,providing control over the relative phases of the state of a photonspread over multiple modes. Examples of transfer matrices that definesuch a phase shifts are given by (for applying a +1 and −i phase shiftto the second mode, respectively):

$\begin{matrix}{s = \begin{pmatrix}1 & 0 \\0 & i\end{pmatrix}} & (14)\end{matrix}$ $s^{\dagger} = \begin{pmatrix}1 & 0 \\0 & {- i}\end{pmatrix}$

For silica-on-silicon materials some embodiments implement variablephase-shifters using thermo-optical switches. The thermo-opticalswitches use resistive elements fabricated on the surface of the chip,that via the thermo-optical effect can provide a change of therefractive index n by raising the temperature of the waveguide by anamount of the order of 10⁻⁵ K. One of skill in the art with access tothe present disclosure will understand that any effect that changes therefractive index of a portion of the waveguide can be used to generate avariable, electrically tunable, phase shift. For example, someembodiments use beam splitters based on any material that supports anelectro-optic effect, so-called χ² and χ³ materials such as lithiumniobite, BBO, KTP, and the like and even doped semiconductors such assilicon, germanium, and the like.

Beam-splitters with variable transmissivity and arbitrary phaserelationships between output modes can also be achieved by combiningdirectional couplers and variable phase-shifters in a Mach-ZehnderInterferometer (MZI) configuration 300, e.g., as shown in FIG. 3A.Complete control over the relative phase and amplitude of the two modes302 a, 302 b in dual rail encoding can be achieved by varying the phasesimparted by phase shifters 306 a, 306 b, and 306 c and the length andproximity of coupling regions 304 a and 304 b. FIG. 3B shows a slightlysimpler example of a MZI 310 that allows for a variable transmissivitybetween modes 302 a, 302 b by varying the phase imparted by the phaseshifter 306. FIGS. 3A and 3B are examples of how one could implement amode coupler in a physical device, but any type of mode coupler/beamsplitter can be used without departing from the scope of the presentdisclosure.

In some embodiments, beam splitters and phase shifters can be employedin combination to implement a variety of transfer matrices. For example,FIG. 4A shows, in a schematic form similar to that of FIG. 2A, a modecoupler 400 implementing the following transfer matrix:

$\begin{matrix}{T_{r} = {\frac{1}{\sqrt{2}}{\begin{pmatrix}1 & 1 \\1 & {- 1}\end{pmatrix}.}}} & (15)\end{matrix}$

Thus, mode coupler 400 applies the following mappings:

$\begin{matrix}\left. \left. {❘10} \right\rangle\mapsto\frac{\left. {\left. {❘10} \right\rangle + {❘01}} \right\rangle}{\sqrt{2}} \right. & (16)\end{matrix}$$\left. \left. {❘01} \right\rangle\mapsto\frac{\left. {\left. {❘10} \right\rangle - {❘01}} \right\rangle}{\sqrt{2}} \right.$$\left. \left. {\left. \left. {❘11} \right\rangle\mapsto{\frac{1}{2}\left( {❘20} \right.} \right\rangle + {❘02}} \right\rangle \right).$

The transfer matrix T_(r) of Eq. (15) is related to the transfer matrixT of Eq. (9) by a phase shift on the second mode. This is schematicallyillustrated in FIG. 4A by the closed node 407 where mode coupler 416couples to the first mode (line 212) and open node 408 where modecoupler 416 couples to the second mode (line 214). More specifically,T_(r)=sTs, and, as shown at the right-hand side of FIG. 4A, mode coupler416 can be implemented using mode coupler 216 (as described above), witha preceding and following phase shift (denoted by open squares 418 a,418 b). Thus, the transfer matrix T_(r) can be implemented by thephysical beam splitter shown in FIG. 4B, where the open trianglesrepresent +i phase shifters.

Similarly, networks of mode couplers and phase shifters can be used toimplement couplings among more than two modes. For example, FIG. 5 showsa four-mode coupling scheme that implements a “spreader,” or“mode-information erasure,” transformation on four modes, i.e., it takesa photon in any one of the input modes and delocalizes the photonamongst each of the four output modes such that the photon has equalprobability of being detected in any one of the four output modes. (Thewell-known Hadamard transformation is one example of a spreadertransformation that can be applied to a set of 2^(q) modes for integerq.) As in FIG. 2A, the horizontal lines 512-515 correspond to modes, andthe mode coupling is indicated by a vertical line 516 with nodes (dots)to identify the modes being coupled. In this case, four modes arecoupled. Circuit notation 502 is an equivalent representation to circuitdiagram 504, which is a network of first-order mode couplings. Moregenerally, where a higher-order mode coupling can be implemented as anetwork of first-order mode couplings, a circuit notation similar tonotation 502 (with an appropriate number of modes) may be used.

FIG. 6 illustrates an example optical device 600 that can implement thefour-mode mode-spreading transform shown schematically in FIG. 5 inaccordance with some embodiments. Optical device 600 includes a firstset of optical waveguides 601, 603 formed in a first layer of material(represented by solid lines in FIG. 6 ) and a second set of opticalwaveguides 605, 607 formed in a second layer of material that isdistinct and separate from the first layer of material (represented bydashed lines in FIG. 6 ). The second layer of material and the firstlayer of material are located at different heights on a substrate. Oneof ordinary skill will appreciate that an interferometer such as thatshown in FIG. 6 could be implemented in a single layer if appropriatelow loss waveguide crossing were employed.

At least one optical waveguide 601, 603 of the first set of opticalwaveguides is coupled with an optical waveguide 605, 607 of the secondset of optical waveguides with any type of suitable optical coupler,e.g., the directional couplers described herein (e.g., the opticalcouplers shown in FIGS. 2B, 3A, 3B). For example, the optical deviceshown in FIG. 6 includes four optical couplers 618, 620, 622, and 624.Each optical coupler can have a coupling region in which two waveguidespropagate in parallel. Although the two waveguides are illustrated inFIG. 6 as being offset from each other in the coupling region, the twowaveguides may be positioned directly above and below each other in thecoupling region without offset. In some embodiments, one or more of theoptical couplers 618, 620, 622, and 624 are configured to have acoupling efficiency of approximately 50% between the two waveguides(e.g., a coupling efficiency between 49% and 51%, a coupling efficiencybetween 49.9% and 50.1%, a coupling efficiency between 49.99% and50.01%, and a coupling efficiency of 50%, etc.). For example, the lengthof the two waveguides, the refractive indices of the two waveguides, thewidths and heights of the two waveguides, the refractive index of thematerial located between two waveguides, and the distance between thetwo waveguides are selected to provide the coupling efficiency of 50%between the two waveguides. This allows the optical coupler to operatelike a 50/50 beam splitter.

In addition, the optical device shown in FIG. 6 can include twointer-layer optical couplers 614 and 616. Optical coupler 614 allowstransfer of light propagating in a waveguide on the first layer ofmaterial to a waveguide on the second layer of material, and opticalcoupler 616 allows transfer of light propagating in a waveguide on thesecond layer of material to a waveguide on the first layer of material.The optical couplers 614 and 616 allow optical waveguides located in atleast two different layers to be used in a multi-channel opticalcoupler, which, in turn, enables a compact multi-channel opticalcoupler.

Furthermore, the optical device shown in FIG. 6 includes a non-couplingwaveguide crossing region 626. In some implementations, the twowaveguides (603 and 605 in this example) cross each other without havinga parallel coupling region present at the crossing in the non-couplingwaveguide crossing region 626 (e.g., the waveguides can be two straightwaveguides that cross each other at a nearly 90-degree angle).

Those skilled in the art will understand that the foregoing examples areillustrative and that photonic circuits using beam splitters and/orphase shifters can be used to implement many different transfermatrices, including transfer matrices for real and imaginary Hadamardtransforms of any order, discrete Fourier transforms, and the like. Oneclass of photonic circuits, referred to herein as “spreader” or“mode-information erasure (MIE)” circuits, has the property that if theinput is a single photon localized in one input mode, the circuitdelocalizes the photon amongst each of a number of output modes suchthat the photon has equal probability of being detected in any one ofthe output modes. Examples of spreader or MIE circuits include circuitsimplementing Hadamard transfer matrices. (It is to be understood thatspreader or MIE circuits may receive an input that is not a singlephoton localized in one input mode, and the behavior of the circuit insuch cases depends on the particular transfer matrix implemented.) Inother instances, photonic circuits can implement other transfermatrices, including transfer matrices that, for a single photon in oneinput mode, provide unequal probability of detecting the photon indifferent output modes.

In some embodiments, entangled states of multiple photonic qubits can becreated by coupling (spatial) modes of two (or more) qubits andperforming measurements on other modes. By way of example, FIG. 7 showsa circuit diagram for a Bell state generator 700 that can be used insome dual-rail-encoded photonic embodiments. In this example, waveguides(or modes) 732-1 through 732-4 are initially each occupied by a photon(indicated by a wavy line); waveguides (or modes) 732-5 through 732-8are initially vacuum (unoccupied) modes. (Those skilled in the art willappreciate that other combinations of occupied and unoccupied modes canbe used.)

A first-order mode coupling (e.g., implementing transfer matrix T of Eq.(9)) is performed on pairs of occupied and unoccupied modes as shown bymode couplers 731-1-731-4, with each mode coupler 731 having one inputwaveguide receiving a photon and one input waveguide receiving vacuum.Mode couplers 731 can be, e.g., 50/50 beam splitters so that, forexample, a photon entering on waveguide 732-1 (or a photon entering onwaveguide 732-5) has a 50% probability of emerging on either output ofmode coupler 731-1. In the following description, mode couplers 731 mayalso be referred to as “directional couplers.” Thereafter, amode-information erasure coupling (e.g., implementing a four-mode modespreading transform as shown in FIG. 5 or a second-order Hadamardtransfer matrix) is performed on one output mode of each directionalcoupler 731 (in this example, waveguides 733-5 through 733-8 provideinputs to the mode-information erasure coupling), as shown by modecoupler 737. In the following description, mode coupler 737 may also bereferred to as a “mode coupler network” or “Hadamard network.”Waveguides 733-5 through 733-8 act as “heralding” modes that aremeasured and used to determine whether a Bell state was successfullygenerated on the four output waveguides 733-1 through 733-4. Forinstance, detectors 738-1 through 738-4 can be coupled to the waveguides733-5 through 733-8 after second-order mode coupler 737. Each detector738-1 through 738-4 can output a classical data signal (e.g., a voltagelevel on a conductor) indicating whether it detected a photon (or thenumber of photons detected). These outputs can be coupled to classicaldecision logic circuit 740, which determines whether a Bell state ispresent on the other four waveguides 733-1 through 733-4. For example,decision logic circuit 740 can be configured such that a Bell state isconfirmed (also referred to as “success” of the Bell state generator) ifand only if a single photon was detected by each of exactly two ofdetectors 738-1 through 738-4. In some embodiments, output modes (orwaveguides) 733-1 through 733-4 can be mapped to the logical states oftwo qubits (Qubit 1 and Qubit 2), as indicated in FIG. 7 . Specifically,in this example, the logical state of Qubit 1 is based on occupancy ofmodes 733-1 and 733-2, and the logical state of Qubit 2 is based onoccupancy of modes 733-3 and 733-4. It should be noted that generationof a Bell state by Bell state generator 700 is a non-deterministic (orstochastic) process; that is, inputting four photons as shown does notguarantee that a Bell state will be created on modes 733-1 through733-4. In one implementation, the probability of success is 4/32; inanother implementation, the success probability is 3/16. It should alsobe noted that there are six detection patterns with one photon in eachof two of detectors 738, and that Bell state generator 700 can beexpected to produce a Bell state in all six possible arrangements of thefour output modes. For a given choice of assignment of modes todual-rail qubits (e.g., as shown in FIG. 7 ), Bell state generator 700can produce any of the four two-qubit Bell states defined in Eqs.(3)-(6) above, as well as a “non-qubit” maximally entangled state.Different detection patterns at detectors 738 can correspond todifferent types of Bell states being produced. In some embodiments,based on the particular detection pattern at detectors 738, mode swapscan be selectably applied to modes 733 in order to cast the Bell stateinto a particular type (e.g., a particular one of the four two-qubitBell states defined above). In some embodiments, the mode swap can besubsumed into subsequent operations without the need for active opticalswitches to implement selectable mode swapping at the output of Bellstate generator 700.

In some embodiments, it is desirable to form cluster states of multipleentangled qubits (typically 3 or more qubits, although the Bell statecan be understood as a cluster state of two qubits). One technique forforming larger entangled systems is through the use of an entanglingmeasurement, which is a projective measurement that can be employed tocreate entanglement between systems of qubits. As used herein, “fusion”(or “a fusion operation” or “fusing”) refers to a two-qubit entanglingmeasurement. A “fusion gate” is a structure that receives two inputqubits, each of which is typically part of an entangled system. Thefusion gate performs a projective measurement operation on the inputqubits that produces either one (“type I fusion”) or zero (“type IIfusion”) output qubits in a manner such that the initial two entangledsystems are fused into a single entangled system. Fusion gates arespecific examples of a general class of two-qubit entanglingmeasurements and are particularly suited for photonic architectures.Examples of type I and type II fusion gates will now be described.

FIG. 8A shows a circuit diagram illustrating a type I fusion gate 800 inaccordance with some embodiments. The diagram shown in FIG. 8A isschematic with each horizontal line representing a mode of a quantumsystem, e.g., a photon. In a dual-rail encoding, each pair of modesrepresents a qubit. In a photonic implementation of the gate the modesin diagrams such as that shown in FIG. 8A can be physically realizedusing single photons in photonic waveguides. Most generally, a type Ifusion gate like that shown in FIG. 8A takes qubit A (physicallyrealized, e.g., by photon modes 843 and 845) and qubit B (physicallyrealized, e.g., by photon modes 847 and 849) as input and outputs asingle “fused” qubit that inherits the entanglement with other qubitsthat were previously entangled with either (or both) of input qubit A orinput qubit B.

For example, FIG. 8B shows the result of type-I fusing of two qubits Aand B that are each, respectively, a qubit located at the end (i.e., aleaf) of some longer entangled cluster state (only a portion of which isshown). The qubit 857 that remains after the fusion operation inheritsthe entangling bonds from the original qubits A and B thereby creating alarger linear cluster state. FIG. 8B also shows the result of type-Ifusing of two qubits A and B that are each, respectively, an internalqubit that belongs to some longer entangled cluster of qubits (only aportion of which is shown). As before, the qubit 859 that remains afterfusion inherits the entangling bonds from the original qubits A and Bthereby creating a fused cluster state. In this case, the qubit thatremains after the fusion operation is entangled with the larger clusterby way of four other nearest neighbor qubits as shown.

Returning to the schematic illustration of type I fusion gate 800 shownin FIG. 8A, qubit A is dual-rail encoded by modes 843 and 845, and qubitB is dual-rail encoded by modes 847 and 849. For example, in the case ofpath-encoded photonic qubits, the logical zero state of qubit A (denoted|0

_(A)) occurs when mode 843 is a photonic waveguide that includes asingle photon and mode 845 is a photonic waveguide that includes zerophotons (and likewise for qubit B). Thus, type I fusion gate 800 cantake as input two dual-rail-encoded photon qubits thereby resulting in atotal of four input modes (e.g., modes 843, 845, 847, and 849). Toaccomplish the fusion operation, a mode coupler (e.g., 50/50 beamsplitter) 853 is applied between a mode of each of the input qubits,e.g., between mode 843 and mode 849 before performing a detectionoperation on both modes using photon detectors 855 (which includes twodistinct photon detectors coupled to modes 843 and 849 respectively). Inaddition, to ensure that the output modes are adjacently positioned, amode swap operation 851 can be applied that swaps the position of thesecond mode of qubit A (mode 845) with the position the second mode ofqubit B (mode 849). In some embodiments, mode swapping can beaccomplished through a physical waveguide crossing as described above orby one or more photonic switches or by any other type of physical modeswap.

FIG. 8A shows only an example arrangement for a type I fusion gate andone of ordinary skill will appreciate that the position of the modecoupler and the presence of the mode swap region 851 can be alteredwithout departing from the scope of the present disclosure. For example,beam splitter 853 can be applied between modes 845 and 847. Mode swapsare optional and are not necessary if qubits having non-adjacent modescan be dealt with, e.g., by tracking which modes belong to which qubitsby storing this information in a classical memory.

Type I fusion gate 800 is a nondeterministic gate, i.e., the fusionoperation succeeds with a certain probability less than 1, and in othercases the quantum state that results is not a larger cluster state thatcomprises the original cluster states fused together to a larger clusterstate. More specifically, gate 800 “succeeds,” with probability 50%,when only one photon is detected by detectors 855, and “fails” if zeroor two photons are detected by detectors 855. When the gate succeeds,the two cluster states that qubits A and B were a part of become fusedinto a single larger cluster state with a fused qubit remaining as thequbit that links the two previously unlinked cluster states (see, e.g.,FIG. 8B). However, when the fusion gate fails, it has the effect ofremoving both qubits from the original cluster resource states withoutgenerating a larger fused state.

FIG. 9A shows a circuit diagram illustrating a type II fusion gate 900in accordance with some embodiments. Like other diagrams herein, thediagram shown in FIG. 9A is schematic with each horizontal linerepresenting a mode of a quantum system, e.g., a photon. In a dual-railencoding, each pair of modes represents a qubit. In a photonicimplementation of the gate the modes in diagrams such as that shown inFIG. 9A can be physically realized using single photons in photonicwaveguides. Most generally, a type II fusion gate such as gate 900 takesqubit A (physically realized, e.g., by photon modes 943 and 945) andqubit B (physically realized, e.g., by photon modes 947 and 949) asinput and outputs a quantum state that inherits the entanglement withother qubits that were previously entangled with either (or both) ofinput qubit A or input qubit B. (For type II fusion, if the inputquantum state had N qubits, the output quantum state has N−2 qubits.This is different from type I fusion where an input quantum state of Nqubits leads to an output quantum state having N−1 qubits.)

For example, FIG. 9B shows the result of type-II fusing of two qubits Aand B that are each, respectively, a qubit located at the end (i.e., aleaf) of some longer entangled cluster state (only a portion of which isshown). The resulting qubit system 971 inherits the entangling bondsfrom qubits A and B thereby creating a larger linear cluster state.

Returning to the schematic illustration of type II fusion gate 900 shownin FIG. 9A, qubit A is dual-rail encoded by modes 943 and 945, and qubitB is dual-rail encoded by modes 947 and 949. For example, in the case ofpath encoded photonic qubits, the logical zero state of qubit A (denoted|0

_(A)) occurs when mode 943 is a photonic waveguide that includes asingle photon and mode 945 is a photonic waveguide that includes zerophotons (and likewise for qubit B). Thus, type II fusion gate 900 takesas input two dual-rail-encoded photon qubits thereby resulting in atotal of four input modes (e.g., modes 943, 945, 947, and 949). Toaccomplish the fusion operation, a first mode coupler (e.g., 50/50 beamsplitter) 953 is applied between a mode of each of the input qubits,e.g., between mode 943 and mode 949, and a second mode coupler (e.g.,50/50 beam splitter) 955 is applied between the other modes of each ofthe input qubits, e.g., between modes 945 and 947. A detection operationis performed on all four modes using photon detectors 957(1)-957(4). Insome embodiments, mode swap operations (not shown in FIG. 9A) can beperformed to place modes in adjacent positions prior to mode coupling.In some embodiments, mode swapping can be accomplished through aphysical waveguide crossing as described above or by one or morephotonic switches or by any other type of physical mode swap. Mode swapsare optional and are not necessary if qubits having non-adjacent modescan be dealt with, e.g., by tracking which modes belong to which qubitsby storing this information in a classical memory.

FIG. 9A shows only an example arrangement for the type II fusion gateand one of ordinary skill will appreciate that the positions of the modecouplers and the presence or absence of mode swap regions can be alteredwithout departing from the scope of the present disclosure.

The type II fusion gate shown in FIG. 9A is a nondeterministic gate,i.e., the fusion operation succeeds with a certain probability less than1, and in other cases the quantum state that results is not a largercluster state that comprises the original cluster states fused togetherto a larger cluster state. More specifically, the gate “succeeds” in thecase where one photon is detected by one of detectors 957(1) and 957(4)and one photon is detected by one of detectors 957(2) and 957(3); in allother cases, the gate “fails.” When the gate succeeds, the two clusterstates that qubits A and B were a part of become fused into a singlelarger cluster state; unlike type-I fusion, no fused qubit remains(compare FIG. 8B and FIG. 9B). When the fusion gate fails, it has theeffect of removing both qubits from the original cluster resource stateswithout generating a larger fused state.

FIG. 10 illustrates an example of a qubit entangling system 1001 inaccordance with some embodiments. Such a system can be used to generatequbits (e.g., photons) in an entangled state (e.g., a GHZ state, Bellpair, and the like), in accordance with some embodiments.

In an illustrative photonic architecture, qubit entangling system 1001can include a photon source module 1005 that is optically connected toentangled state generator 1000. Both the photon source module 1005 andthe entangled state generator 1000 may be coupled to a classicalprocessing system 1003 such that the classical processing system 1003can communicate and/or control (e.g., via the classical informationchannels 1030 a-b) the photon source module 1005 and/or the entangledstate generator 1000. Photon source module 1005 may include a collectionof single-photon sources that can provide output photons to entangledstate generator 1000 by way of interconnecting waveguides 1032.Entangled state generator 1000 may receive the output photons andconvert them to one or more entangled photonic states and then outputthese entangled photonic states into output waveguides 1040. In someembodiments, output waveguide 1040 can be coupled to some downstreamcircuit that may use the entangled states for performing a quantumcomputation. For example, the entangled states generated by theentangled state generator 1000 may be used as resources for a downstreamquantum optical circuit (not shown).

In some embodiments, system 1001 may include classical channels 1030(e.g., classical channels 1030-a through 1030-d) for interconnecting andproviding classical information between components. It should be notedthat classical channels 1030-a through 1030-d need not all be the same.For example, classical channel 1030-a through 1030-c may comprise abi-directional communication bus carrying one or more reference signals,e.g., one or more clock signals, one or more control signals, or anyother signal that carries classical information, e.g., heraldingsignals, photon detector readout signals, and the like.

In some embodiments, qubit entangling system 1001 includes the classicalcomputer system 1003 that communicates with and/or controls the photonsource module 1005 and/or the entangled state generator 1000. Forexample, in some embodiments, classical computer system 1003 can be usedto configure one or more circuits, e.g., using system clock that may beprovided to photon sources 1005 and entangled state generator 1000 aswell as any downstream quantum photonic circuits used for performingquantum computation. In some embodiments, the quantum photonic circuitscan include optical circuits, electrical circuits, or any other types ofcircuits. In some embodiments, classical computer system 1003 includesmemory 1004, one or more processor(s) 1002, a power supply, aninput/output (I/O) subsystem, and a communication bus or interconnectingthese components. The processor(s) 1002 may execute modules, programs,and/or instructions stored in memory 1004 and thereby perform processingoperations.

In some embodiments, memory 1004 stores one or more programs (e.g., setsof instructions) and/or data structures. For example, in someembodiments, entangled state generator 1000 can attempt to produce anentangled state over successive stages, any one of which may besuccessful in producing an entangled state. In some embodiments, memory1004 stores one or more programs for determining whether a respectivestage was successful and configuring the entangled state generator 1000accordingly (e.g., by configuring entangled state generator 1000 toswitch the photons to an output if the stage was successful, or pass thephotons to the next stage of the entangled state generator 1000 if thestage was not yet successful). To that end, in some embodiments, memory1004 stores detection patterns (described below) from which theclassical computing system 1003 may determine whether a stage wassuccessful. In addition, memory 1004 can store settings that areprovided to the various configurable components (e.g., switches)described herein that are configured by, e.g., setting one or more phaseshifts for the component.

In some embodiments, some or all of the above-described functions may beimplemented with hardware circuits on photon source module 1005 and/orentangled state generator 1000. For example, in some embodiments, photonsource module 1005 includes one or more controllers 1007-a (e.g., logiccontrollers) (e.g., which may comprise field programmable gate arrays(FPGAs), application specific integrated circuits (ASICS), a “system ona chip” that includes classical processors and memory, or the like). Insome embodiments, controller 1007-a determines whether photon sourcemodule 1005 was successful (e.g., for a given attempt on a given clockcycle, described below) and outputs a reference signal indicatingwhether photon source module 1005 was successful. For example, in someembodiments, controller 1007-a outputs a logical high value to classicalchannel 1030-a and/or classical channel 1030-c when photon source module1005 is successful and outputs a logical low value to classical channel1030-a and/or classical channel 1030-c when photon source module 1005 isnot successful. In some embodiments, the output of control 1007-a may beused to configure hardware in controller 1007-b.

Similarly, in some embodiments, entangled state generator 1000 includesone or more controllers 1007-b (e.g., logical controllers) (e.g., whichmay comprise field programmable gate arrays (FPGAs), applicationspecific integrated circuits (ASICS), or the like) that determinewhether a respective stage of entangled state generator 1000 hassucceeded, perform the switching logic described above, and output areference signal to classical channels 1030-b and/or 1030-d to informother components as to whether the entangled state generator 400 hassucceeded.

In some embodiments, a system clock signal can be provided to photonsource module 1005 and entangled state generator 1000 via an externalsource (not shown) or by classical computing system 1003 generates viaclassical channels 1030-a and/or 1030-b. In some embodiments, the systemclock signal provided to photon source module 1005 triggers photonsource module 1005 to attempt to output one photon per waveguide. Insome embodiments, the system clock signal provided to entangled stategenerator 1000 triggers, or gates, sets of detectors in entangled stategenerator 1000 to attempt to detect photons. For example, in someembodiments, triggering a set of detectors in entangled state generator1000 to attempt to detect photons includes gating the set of detectors.

It should be noted that, in some embodiments, photon source module 1005and entangled state generator 1000 may have internal clocks. Forexample, photon source module 1005 may have an internal clock generatedand/or used by controller 1007-a and entangled state generator 1000 hasan internal clock generated and/or used by controller 1007-b. In someembodiments, the internal clock of photon source module 1005 and/orentangled state generator 1000 is synchronized to an external clock(e.g., the system clock provided by classical computer system 1003)(e.g., through a phase-locked loop). In some embodiments, any of theinternal clocks may themselves be used as the system clock, e.g., aninternal clock of the photon source may be distributed to othercomponents in the system and used as the master/system clock.

In some embodiments, photon source module 1005 includes a plurality ofprobabilistic photon sources that may be spatially and/or temporallymultiplexed, i.e., a so-called multiplexed single photon source. In oneexample of such a source, the source is driven by a pump, e.g., a lightpulse, that is coupled into an optical resonator that, through somenonlinear process (e.g., spontaneous four wave mixing, second harmonicgeneration, and the like) may generate zero, one, or more photons. Asused herein, the term “attempt” is used to refer to the act of driving aphoton source with some sort of driving signal, e.g., a pump pulse, thatmay produce output photons non-deterministically (i.e., in response tothe driving signal, the probability that the photon source will generateone or more photons may be less than 1). In some embodiments, arespective photon source may be most likely to, on a respective attempt,produce zero photons (e.g., there may be a 90% probability of producingzero photons per attempt to produce a single-photon). The second mostlikely result for an attempt may be production of a single-photon (e.g.,there may be a 9% probability of producing a single-photon per attemptto produce a single-photon). The third most likely result for an attemptmay be production of two photons (e.g., there may be an approximately 1%probability of producing two photons per attempt to produce a singlephoton). In some circumstances, there may be less than a 1% probabilityof producing more than two photons.

In some embodiments, the apparent efficiency of the photon sources maybe increased by using a plurality of single-photon sources andmultiplexing the outputs of the plurality of photon sources.

The precise type of photon source used is not critical and any type ofsource can be used, employing any photon generating process, such asspontaneous four wave mixing (SPFW), spontaneous parametricdown-conversion (SPDC), or any other process. Other classes of sourcesthat do not necessarily require a nonlinear material can also beemployed, such as those that employ atomic and/or artificial atomicsystems, e.g., quantum dot sources, color centers in crystals, and thelike. In some cases, sources may or may be coupled to photonic cavities,e.g., as can be the case for artificial atomic systems such as quantumdots coupled to cavities. Other types of photon sources also exist forSPWM and SPDC, such as optomechanical systems and the like. In someexamples the photon sources can emit multiple photons already in anentangled state in which case the entangled state generator 400 may notbe necessary, or alternatively may take the entangled states as inputand generate even larger entangled states.

For the sake of illustration, an example which employs spatialmultiplexing of several non-deterministic is described as an example ofa mux photon source. However, many different spatial mux architecturesare possible without departing from the scope of the present disclosure.Temporal muxing can also be implemented instead of or in combinationwith spatial multiplexing. mux schemes that employ log-tree, generalizedMach-Zehnder interferometers, multimode interferometers, chainedsources, chained sources with dump-the-pump schemes, asymmetricmulti-crystal single photon sources, or any other type of muxarchitecture can be used. In some embodiments, the photon source canemploy a mux scheme with quantum feedback control and the like.

The foregoing description provides an example of how photonic circuitscan be used to implement physical qubits and operations on physicalqubits using mode coupling between waveguides. In these examples, a pairof modes can be used to represent each physical qubit. Examplesdescribed below can be implemented using similar photonic circuitelements.

The following sections describe examples of optical circuits andmultiplexing techniques that can be used to spatially (and temporally)align photons. Such circuits and techniques can be applied in a widevariety of photonic systems and circuits.

2. Spatial Multiplexing of Photons

If photons can be reliably generated on demand (e.g., in response topump pulses as described above), multiple photons can be providedsimultaneously to a circuit such as Bell state generator 700 simply byproviding an appropriate number of photon sources (four in the case ofBell state generator 700) and pumping (or otherwise triggering) all ofthe photon sources simultaneously. However, as described above, knownsingle-photon sources operate non-deterministically, and a given photonsource may or may not produce a photon pair in response to a given pumppulse. If, for example, four non-deterministic photon sources are usedto provide photons to input waveguides 732-1 through 732-4 of Bell stategenerator 700, even if all four sources are pumped for each time bin,the probability of four photons arriving on input waveguides 732-1through 732-4 in any given time bin would be less than 1.

One technique to improve the likelihood of simultaneously obtainingphotons from each of a set of non-deterministic photon sources involvesspatial multiplexing of multiple photon sources. FIG. 11 shows anexample of an N×1 spatial multiplexing circuit 1100 for a set of Nphoton sources 1102-1 through 1102-N for some number N, where N≥2. Eachphoton source 1102 is a different physical device that can produce aphoton pair in response to a pump pulse. For instance, each photonsource 1102 can be a heralded single photon source as described above.Photon sources 1102 can be pumped repeatedly, and each instance ofpumping photon sources 1102 can define a time bin (or temporal mode).For each time bin, each photon source 1102 might or might not produce aphoton pair. Each photon source 1102 has an associated detector 1104 andan associated signaling waveguide 1122. In any time bin where aparticular photon source 1102 does produce a photon pair, one photonpropagates through the associated signaling waveguide 1122 while theother photon is detected by the associated detector 1104.

In each time bin, each photon source 1102 might or might not generate aphoton. Dots 1106 a-1106 f show an example of photons that might begenerated during different time bins P1-P5. FIG. 11 can be regarded as asnapshot view, with photons 1106 produced during different time binsappearing at different locations along the waveguides 1122.

An N×1 multiplexer (or “mux”) 1120 can be an active optical switchingcircuit that selectably couples one of N input waveguides 1134 to anoutput waveguide 1136, and selectable optical coupling can be providedusing active optical switches or other active optical components thatcan be controlled to either allow or block propagation of photons. Forexample, N×1 mux 1120 can be implemented as an N×1 generalizedMach-Zehnder interferometer (GMZI). An N×M GMZI is an optical circuitthat can receive photons on a set of N input waveguides and control aset of active phase shifters to selectably couple M of the receivedphotons to a set of M output waveguides. (In the case of mux 1120, M=1.)Additional description of GMZI implementations can be found below. N×1mux 1120 can be controlled by control logic 1130, which can be aconventional electronic logic circuit. Control logic 1130 can receivesignals from each of detectors 1104 that indicate, for each time bin,whether a photon was or was not detected by each detector 1104.Accordingly, control logic 1130 can determine which photon sources 1102produced photons during a given time bin (and therefore which inputwaveguides 1134 are carrying photons for that time bin). For each timebin, control logic 1130 can control N×1 mux 1120 to couple one inputwaveguide that has a photon to output waveguide 1136. For example, aGMZI includes a set of active phase shifters that can be controlled toapply variable phase shifts along different optical paths, creatingeither constructive or destructive interference, and control logic 1130can generate control signals to set the state of each active phaseshifter in a GMZI implementing N×1 mux 1120 to provide the desiredcoupling.

The time bin can be as long or short as desired, based oncharacteristics of the optical circuit, variability in the timing ofgenerating photons in single photon sources 1102, etc. In someinstances, an interval between time bins may be determined based on thespeed at which N×1 mux 1120 can be switched, on a recovery time forphoton sources 1102 and/or detectors 1104, operating speed of circuitsdownstream of N×1 mux 1120, or other design considerations to allow eachtime bin to be treated as an independent temporal mode.

As noted above, the behavior of photon sources 1102 may benon-deterministic. That is, during a given time bin, the probability ofa photon being generated by a given photon source 1102 can berepresented as p_(s), where p_(s)<1. For photon sources of this type,multiplexing as shown in FIG. 11 provides the ability to increase theprobability of successfully producing a photon in a given time bin. Asshown in FIG. 11 , if N non-deterministic single-photon sources areused, with one photon source coupled to each input of N×1 mux 1120, andif each photon source has probability p_(s) of generating a photon (fora given time bin), then the probability that N×1 mux 1120 receives atleast one photon is p_(mux)=1−(1−p_(s))^(N). Thus, for a given type ofphoton source 1102, a desired probability p_(mux) of providing onephoton per time bin to output waveguide 1136 can, at least in principle,be achieved by a suitable choice of N. (As a practical matter, somecombinations of p_(s) and p_(mux) may require a prohibitively largenumber N of photon sources.)

In some applications, a downstream circuit may require multiple photonsas inputs. For example Bell state generator 700 of FIG. 7 can produce aBell state only if four photons are input simultaneously. Accordingly,to reliably provide four input photons per time bin to Bell stategenerator 700, four instances of circuit 1100 can be provided, with eachinstance having an output 1136 coupled to a different one of inputwaveguides 732-1 through 732-4.

3. Raster Mux Circuits

Providing four instances of circuit 1100 may consume a significantamount of area, especially when N is large. According to someembodiments, circuit area can be reduced using a technique referred toas “raster multiplexing” (or “raster mux” or “rastering”) that uses Ninput photon sources to produce R simultaneous output photons on Routput waveguides. FIG. 12 shows a simplified schematic view of a rastermux circuit 1200 according to some embodiments. Raster mux circuit 1200includes a GMZI 1220 that, for each time bin, selects one of N inputpaths 1222 to optically couple to an output path; however, instead ofjust one output path, GMZI 1220 has R selectable output paths 1236.

Control logic 1230 can be implemented as a digital logic circuit with anarrangement of classical logic gates (AND, OR, NOR, XOR, NAND, NOT,etc.), such as a field programmable gate array (FPGA) orsystem-on-a-chip (SOC) having a programmable processor and memory, or anon-chip hard-wired circuit, such as an application specific integratedcircuit (ASIC). In some embodiments, GMZI 1220 is coupled to an off-chipclassical computer having a processor and a memory, and the off-chipclassical computer is programmed to perform some or all of theoperations of control logic 1230. In some embodiments, control logic1230 (which can include on-chip and/or off-chip components) can beprovided with program code providing decision rules to select controlsignals for GMZI 1220, and control logic 1230 can execute the programcode and generate appropriate control signals.

In operation, for each time bin, control logic 1230 selects one of theinput (spatial) paths 1222 as an active input path to optically coupleto an active one of output paths 1236. Selection of an input path can bebased on signals received by control logic 1230 (indicated by inputarrow 1231) that indicate which of input paths 1222 have a propagatingphoton. For instance, as described above with reference to FIG. 11 ,each photon source 1102 can have an associated detector 1106. Controllogic 1230 can receive heralding signals from detectors 1106 and selectan active input path based on the heralding signals. In addition toselecting an active input path, control logic 1230 selects one of outputpaths 1236 as an active output path on a rotating or cyclic basis. Forexample, for each time bin, control logic 1230 can increment a counterand can select one of output paths 1236 based on the counter value(modulo R). For instance, output path 1236-1 can be selected for a firsttime bin, output path 1236-2 for the next time bin, and so on untiloutput path 1236-R is selected for the Rth time bin. In this manner,raster mux circuit 1200 can produce a set of R photons for a set of Rtime bins, with each photon being output on a different one of the Routput paths 1236 in a different time bin, in a known (controlled)order. A set of R time bins is sometimes referred to herein as a “rasterperiod.”

In some embodiments, the set of R output photons can be synchronized intime by introducing appropriate synchronization delays, as shown in syncdelay circuit 1250. Loops 1232 indicate an amount of delay introduced oneach optical path. For instance, each loop 1232 can indicate one addedtime bin of delay. Delay can be implemented, e.g., by introducingadditional lengths of optical waveguide material or by other techniquesthat lengthen the optical path. In the example shown, sync delay boxadds R−1 time bins of delay to output path 1236-1, R−2 time bins tooutput path 1236-2, and so on until output line 1236-R has no added timebins of delay. Accordingly, the R photons (indicated by dots 1206)output onto different output paths 1236 for successive time bins canarrive simultaneously at the outputs of sync delay circuit 1250. In thismanner, a single instance of raster mux circuit 1200 with sync delaycircuit 1250 can provide a set of R simultaneous photons on Rwaveguides. Raster mux circuit 1200 can be characterized as an “N×Rraster mux circuit,” indicating N inputs and R outputs. It should benoted that if the inputs are provided to raster mux circuit 1200according to a given time bin time t (e.g., a pump pulse period forphoton sources 1102), a set of outputs is generated in time Rt.

Circuit 1200 is illustrative, and variations and modifications arepossible. In some embodiments, GMZI 1220 can be replaced with otheractive switching circuits that can selectably couple one of N inputpaths to one of R output paths. If desired, the output photons can besynchronized by adding appropriate delay to each output path, e.g.,using sync delay circuit 1250.

FIG. 13 shows a flow diagram of a process 1300 that can be implementedin control logic 1230 according to some embodiments. At block 1302,control logic 1230 can receive input signals 1231 indicating which ofthe N input paths 1222 of GMZI 1220 have photons arriving in the currenttime bin. For instance as shown in FIG. 11 , photon sources 1102 canhave associated detectors 1106 that generate signals (e.g., classicaldigital logic signals) indicating whether a photon was detected. Thissignal can be used by control logic 1230 as an indicator of a photon onthe corresponding input path 1222.

At block 1304, control logic 1230 can select an active output path (oneof output paths 1236) based on a cycle counter. For instance, controllogic 1230 can implement a cyclic counter with R values, and the activeoutput path can be selected based on the current value of the cycliccounter. Other selection logic can be used, provided that output paths1236 are selected in a rotating or cyclic order such that each outputpath 1236 is selected once for each group of R consecutive time bins (orraster period). The same selection pattern can be repeated for eachraster period.

At block 1306, control logic 1230 can select an active input path(waveguide) based on the input signals received at block 1302. Forexample, control logic 1230 can select one input path 1222 that isoccupied by a photon (in the current time bin) as an active input path.For time bins where only one input path 1222 has a photon, then controllogic 1230 can select that path as the active path. For time bins wheremultiple input paths 1222 are occupied, control logic 1230 can apply aprioritization rule to select one of the input paths that is occupied.For instance, the input paths can be assigned numbers, and thelowest-numbered input path that is occupied can be selected. Otherprioritization rules can be substituted, as long as only one activeinput path is selected for each time bin. In some embodiments, theprioritization rules can depend in part on which output path is selectedas the active output path at block 1306. (For example, depending on theGMZI implementation, couplings between certain combinations of input andoutput waveguides may have lower loss, or higher efficiency, than othercombinations, and the prioritization rules can favor input/outputcouplings that have higher efficiency.)

At block 1308, control logic 1230 can determine a set of control signalsfor the active phase shifters of GMZI 1220 that will result in theactive input path being coupled to the active output path and otheroutput paths being blocked (coupled to vacuum input paths). In someembodiments, a lookup table can be provided with an entry for eachpairing of active input and output paths, and each entry can include alist of corresponding switch settings for the active phase shifters.Accordingly, at block 1308, control logic 1230 can access the lookuptable and read the switch settings. Other implementations can besubstituted. At block 1310, control logic 1230 can send control signalsto the active switches of GMZI 1220. In some embodiments, sending thecontrol signals can include applying specific voltages to active phaseshifters to control the phase shift.

At block 1312, control logic 1230 can increment the cycle counter. Asprocess 1300 iterates, incrementing the cycle counter results in thenext output path in the rotation being selected as the active outputpath for the next time bin.

Process 1300 is illustrative, and variations and modifications arepossible. Blocks or operations described sequentially can be performedin parallel, and order of operations can be modified to the extent thatlogic permits. Input paths 1222 should have sufficient length that theinput signals indicating path occupancy for a given time bin can bereceived and control signals sent to GMZI 1220 before the photonsassociated with those input signals reach GMZI 1220. In someembodiments, at the end of each raster period, one or more idle timebins can be introduced, e.g., to allow a recovery period for detectorsor other circuit components, before beginning the next raster period.More generally, selection of an output path from a group of output pathscan be based on timing considerations and can be independent of theselection of the active input path. For example, control logic 1230 canmaintain an ordered list of output paths in a raster group, and eachtime control logic 1230 is triggered to select an output path, controllogic 1230 can select the next output path from the list. Selection ofan output path in this manner can but need not occur according to afixed clock cycle or other regular time interval. For instance, in someembodiments control logic 1230 can wait until an input signal indicatingan occupied path is received and select the next output path from thelist in response to the input signal, which may or may not occur atregular time intervals.

In some embodiments, the speed at which raster mux circuit 1200 canoperate may be limited by the speed of various components. For instance,active phase shift circuits in GMZI 1220 may have a maximum switchingspeed, or detectors 1106 that generate signals may experience deadtimeafter detecting a photon. The duration of a time bin can be selected asdesired, provided that it is long enough to allow the optical circuit tooperate correctly. (It should be understood that photons in differenttime bins may be propagating through different components of an opticalcircuit at the same time.)

4. Example Applications of Raster Mux Circuits

4.1. Rasterized Inputs to a Single Downstream Circuit

FIG. 14 shows a simplified schematic view of an optical circuit thatincludes an N×4 raster mux circuit 1420 coupled to a Bell stategenerator 700 according to some embodiments. Bell state generator 700can be implemented as described above with reference to FIG. 7 . Rastermux circuit 1420 can be an implementation of raster mux circuit 1200with R=4. For each time bin, a set of N photon sources 1102 (which canbe heralded single photon sources as described above) can be pumped orotherwise triggered to (non-deterministically) produce photons, andraster mux circuit 1420 can select a photon from any one of the Nsources on to propagate on one of output waveguides 1436. Raster muxcircuit 1420 can also select the output waveguide 1436 on a rotating orcyclic basis as described above. Sync delay circuit 1450 can be similarto sync delay circuit 1250 described above, introducing 3, 2, 1, or zerotime bins of delay to each of output paths 1436. At the end of four timebins, four photons can be delivered simultaneously to input paths 732-1through 732-4 of Bell state generator 700. As compared to providing aseparate N×1 multiplexer 1120 for each input to Bell state generator700, the area required to implement circuit 1400 of FIG. 14 issignificantly reduced. The tradeoff is in throughput: where a set offour N×1 multiplexers can, in principle, produce four photons per timebin, N×4 raster mux circuit 1420 can produce four photons every fourthtime bin. In a different comparison, assuming that the number N ofphoton sources is a limiting factor, a circuit having a separate (N/4)×1multiplexer 1120 for each input to Bell state generator 700 results in acircuit area similar to that occupied by circuit 1400; however, forexisting single-photon sources and currently practical values of N, theprobability of obtaining four photons in the same time bin from four(N/4)×1 multiplexers is lower than the probability of obtaining fourphotons in the same time bin from four N×1 multiplexers. Consequently,despite the reduced speed, circuit 1400 with a single N×4 raster mux1420 can produce Bell states at a comparable or even higher rate than acircuit using separate (N/4)×1 multiplexers for each input to Bell stategenerator 700.

In some embodiments, the speed/area tradeoff can be optimized by usingmultiple raster mux circuits with each raster mux circuit producing morethan one but fewer than all of the input photons for a downstreamcircuit element. As an example FIG. 15 shows a simplified schematic viewof an optical circuit 1500 that includes two (N/2)×2 raster mux circuits1520 coupled to a Bell state generator 700 according to someembodiments. Bell state generator 700 can be implemented as describedabove with reference to FIG. 7 . Each raster mux circuit 1520 can be animplementation of raster mux circuit 1200 with R=2, and each raster muxcircuit 1520 can be coupled to a different set of N/2 photon sources1102.

All N photon sources 1102 can be operated on each time bin to producephotons, and each raster mux circuit 1520 can select a photon from oneof its (N/2) sources on each time bin to propagate on one of outputwaveguides 1536. Each raster mux circuit 1500 can also select the outputwaveguide 1536 on a rotating (in this case alternating) basis asdescribed above. Sync delays 1550 can delay one output of each rastermux 1500 relative to the other output of the same raster mux 1500. Atthe end of two time bins, four photons can be delivered simultaneouslyto input paths 732-1 through 732-4 of Bell state generator 700: two fromraster mux circuit 1500-1 and two from raster mux circuit 1500-2.

Circuit 1500 of FIG. 15 uses a similar area (for the same value of N) tocircuit 1400 of FIG. 14 , and circuit 1500 can provide inputs to Bellstate generator 700 at twice the rate of circuit 1400. In someembodiments, due to the increased speed, the circuit of FIG. 15 canobtain comparable throughput (measured in average number of four-photongroups per time period) to the circuit of FIG. 14 using only N′=N/2inputs to each raster mux circuit 1500. Thus, the circuit of FIG. 15 cangive comparable performance to the circuit of FIG. 14 while consumingsimilar area.

In circuits 1400 and 1500 of FIGS. 14 and 15 , raster multiplexing isused to provide input photons to a Bell state generator. In variousembodiments, raster multiplexing can be used in a similar manner toprovide multiple photons to any downstream circuit. FIGS. 16A-16C showexamples of how a raster mux circuit can be used to enable a single copyof an “upstream” circuit to provide multiple inputs to a “downstream”circuit according to some embodiments.

Shown in FIG. 16A is a configuration of optical circuits 1600 with threecopies of an upstream circuit 1602 each providing an input to adownstream circuit 1604. Each copy of upstream circuit 1602 can be aninstance of any optical circuit that provides a photon on an outputwaveguide (or in some instances multiple photons on multiplewaveguides). For example, each copy of upstream circuit 1602 can includea set of photon sources coupled to an N×1 multiplexer as described abovewith reference to FIG. 11 . Any other optical circuit, including anoptical circuit that produces a group of photons on different waveguides(rather than a single photon on a single waveguide as in the circuit ofFIG. 11 ) can also be used as upstream circuit 1602. Upstream circuits1602 are all copies of each other, meaning that they include physicallyseparate sets of components that have the same optical characteristicsand couplings. Downstream circuit 1604 can be any optical circuit thatoperates on a set of multiple photons received simultaneously. As shown,downstream circuit 1604 can receive one input (or group of inputs) fromeach copy of upstream circuit 1602. For example, downstream circuit 1604can implement Bell state generator 700 of FIG. 7 . Any other opticalcircuit that operates on multiple inputs (or multiple groups of inputs)received simultaneously can be substituted. In the example shown,downstream circuit 1604 receives inputs from three copies of upstreamcircuit 1602; however, any number of copies (e.g., 2, 4, or more) can beused depending on the particular number of inputs (or groups of inputs)used by downstream circuit 1604. In some embodiments, downstream circuit1604 can provide one or more photons as an output. In addition orinstead, downstream circuit 1604 can consume some or all of the inputphotons (e.g., downstream circuit 1604 can include a detector) andproduce output in another form such as electronic signals from adetector.

FIG. 16B shows a circuit 1620 according to some embodiments thatprovides the same functionality as circuit 1600 of FIG. 16A. Circuit1620 can includes a single copy of upstream circuit 1602, a raster muxcircuit 1622, a synchronization delay unit 1624, and downstream circuit1604. Raster mux circuit 1622 can be an implementation of N×R raster muxcircuit 1200 of FIG. 12 . In this example, N=1 and R=3. (Other sizes canbe substituted, depending on the number of inputs to downstream circuit1604.) Synchronization delay circuit 1624 can implement delays of 2, 1,and 0 time bins on the output lines of raster mux circuit 1622, anddownstream circuit 1604 can receive a set of three simultaneous inputsonce every three time bins. It should be noted that operation ofdownstream circuit 1604 can be agnostic to whether its inputs areprovided using multiple copies of upstream circuit 1602 (as shown inFIG. 16A) or a single copy of upstream circuit 1602 (as shown in FIG.16B). Similarly, operation of upstream circuit 1602 can be agnostic asto whether its outputs are delivered to raster mux circuit 1622 ordirectly to downstream circuit 1604.

In some embodiments, upstream circuit 1602 may already include amultiplexer for output selection. For instance, upstream circuit 1602may generate a number N of possible outputs and include an N×1multiplexer to select one output. In such embodiments, the N×1multiplexer can be replaced by an N×R raster mux circuit. FIG. 16C showsan example in which upstream circuit 1602′ has been modified to includea raster mux circuit 1644 that provides outputs on one of threealternative output paths. Raster mux circuit 1644 in this example can bean N×3 raster mux circuit, where N is the number of alternative outputsfrom which the actual output is selected. More generally, raster muxcircuit 1644 can be an N×R raster mux circuit, where R is the number ofinputs to be provided to downstream circuit 1604. Combining outputselection with raster multiplexing in upstream circuit 1602′ can reducethe number of active optical switches in a given photon path.Synchronization delay unit 1624 can be used to deliver inputssimultaneously to downstream circuit 1604.

Using the principle illustrated in FIGS. 16A-16C, in any optical circuitarrangement where a downstream circuit operates on inputs provided bymultiple copies of an upstream circuit, the multiple copies of theupstream circuit can be replaced by a single copy of the upstreamcircuit with a raster mux circuit and appropriate synchronizationdelays.

4.2.Rasterized Inputs to Multiple Bell State Generators

In embodiments described above, a single raster mux circuit can providemultiple inputs to a downstream circuit. In other embodiments, multipleraster mux circuits can provide inputs to multiple downstream circuits.

By way of example, FIG. 17 shows a simplified schematic diagram of anoptical circuit 1700 according to some embodiments. Circuit 1700includes a number R of Bell state generator (BSG) circuits 1704, each ofwhich can be an instance of Bell state generator 700 described above.Four N×R raster mux circuits 1710 are coupled to the input paths of BSGcircuits 1704 with each raster mux circuit 1710 having one of its Routput paths coupled to an input path of each BSG circuit 1704. Eachraster mux circuit 1710 can be an instance of raster mux circuit 1200and can receive and select among inputs from a group of N single photonsources as described above. In circuit 1700, each raster mux circuit1710 supplies a different one of the four inputs to each BSG circuit1704.

Raster mux circuits 1710 can be operated synchronously such that, duringa first time bin, each raster mux circuit 1710 directs its output to BSGcircuit 1704-1, during a second time bin, each raster mux circuit 1710directs its output to BSG circuit 1704-2, and so on until during an Rthtime bin, each raster mux circuit 1710 directs its output to BSG circuit1704-R. Accordingly, each BSG 1704 can receive all four of its inputphotons simultaneously (in the same time bin) and can(non-deterministically) generate a Bell state output in the mannerdescribed above. Each BSG circuit 1704 generates a Bell state (if itdoes so) during a different time bin. To facilitate downstreamoperations using the outputs of two or more of Bell state generators1704, delay circuits 1720 can be provided. Delay circuit 1720-1 delaysall four outputs of BSG circuit 1704-1 by R−1 time bins, delay circuit1720-2 delays all four outputs of BSG circuit 1704-2 by R−2 time bins,and so on, with delay circuit 1720-R adding zero time bins of delay. Itshould be understood that the added delay is defined relative to otherdelay circuits 1720.

In circuit 1700, each BSG circuit 1704 is “active” (receiving photonsusable to generate a Bell state) for a different one of every set of Rtime bins. Due to the nature of GMZI circuits, in some embodiments, oneor another of raster mux circuits 1710 may occasionally generate an“errant” photon, i.e., a photon on an output path other than the activeoutput path, in addition to a photon on the active output path. In someembodiments, each output path of each raster mux circuit 1710 caninclude a blocking switch 1730 (shown as dashed-line boxes), and thecontrol logic in each raster mux circuit 1710 (e.g., control logic 1230of FIG. 12 ) can set the state of blocking switches 1730 such thatphotons on any output path other than the active output path areblocked. Blocking switches 1730 can each be implemented using anytechnique that results in a photon being selectably blocked or allowedto propagate through a waveguide. For example, a blocking switch can beimplemented using a (2×2) Mach Zehnder interferometer and “dumping” onepath (e.g., by making one waveguide a dead end). As another example, ablocking switch can be implemented by providing dopants in a region ofthe waveguide that cause the photon to be absorbed or not as a functionof an applied voltage. Other implementations may also be used. In someembodiments, blocking switches 1730 can be “normally blocking” such thatphotons are blocked unless a signal (e.g., a voltage) to permit photonpropagation is actively applied. In other embodiments, blocking switches1730 can be “normally open” such that photons propagate unless a signalto block photon propagation is actively applied. Blocking switches canbe implemented with any raster mux circuit in a similar manner.

It will be appreciated that circuit 1700 is illustrative. A set ofraster mux circuits can be used to provide inputs to any set of Rdownstream circuits, not limited to BSG circuits. In general, if each ofthe R downstream circuits uses M inputs, then M copies of an N×R rastermux circuit can be used to provide inputs. (N is the number of inputsfrom which the raster mux circuit selects the output and depending onthe upstream circuit, N can be any number greater than or equal to 1.)In some embodiments, in addition to or instead of blocking switches,clocked electrical gating can be applied to output signals from thedetectors in each BSG circuit 1704, such that signals from the detectorsare ignored except during the time bin when that BSG circuit 1704 isactive. Using these or other techniques, errant photons can be preventedfrom affecting circuit operations or output data.

Circuit 1700 is drawn in a manner that suggests that a raster muxcircuit selects output paths sequentially according to their physicalarrangement. This can be, but need not be, the case, and in variousembodiments, output paths for successive time bins can be selected inany order, as long as each of the R output paths is selected once duringeach raster period. By way of example, FIG. 18 shows a simplifiedschematic view of a circuit 1800 according to some embodiments. Circuit1800 includes an N×6 raster mux circuit 1810, which can be implementedsimilarly to raster mux circuit 1200 or other raster mux circuitsdescribed herein. In this example, raster mux circuit 1810 has one inputpath 1836 coupled to each of R=6 BSG circuits. For example raster muxcircuit 1810 can be one of raster mux circuits 1710 of FIG. 17 . In thisexample, the arrangement of output paths 1836 in the drawing is intendedto represent the relative positions of waveguides. Each output path 1836is labeled with the time bin for which it is active. In this example, async delay unit 1850 is placed downstream of raster mux circuit 1810 andupstream of the BSG circuits, and all BSG circuits can receive theirinputs in the same time bin. In this particular example, the physicalarrangement of output paths 1836 is assumed to correspond to thedrawing; thus FIG. 18 shows an implementation in which adjacent outputpaths 1836 are not selected for successive time bins. Instead, theselection of output paths starts with the center paths 1836-1, 1836-2,and proceeds outward in an alternating fashion. For some GMZIconfigurations, an alternating selection pattern as shown in FIG. 18 canavoid the generation of errant photons on output paths 1836 without theuse of blocking switches. More generally, in some embodiments the orderin which output paths of a raster mux circuit are selected within araster period can be determined based in part on which selectionorder(s) can avoid or minimize generation of errant photons.

4.3. Raster Mux for Single-Qubit and Two-Qubit Measurement Operations

In quantum computing and/or quantum communication applications of linearoptical circuits, it may be desirable to perform measurements on photonsthat encode qubit states. For instance, a pair of waveguides can be usedto encode a qubit using a dual-rail encoding as described above.According to some embodiments, raster multiplexing can be used toprovide input qubits for quantum operations such as fusion operations(as described above) and/or single-qubit measurements. FIGS. 19A and 19Btogether show a simplified circuit schematic of an optical circuit 1900according to some embodiments. Circuit 1900 implements selectable fusionor single-qubit measurement operations on pairs of qubits. Referringfirst to FIG. 19A, circuit 1900 includes a set of N entanglementcircuits 1902. Each entanglement circuit 1902 can be a circuit thatgenerates an entangled system of two or more qubits. Examples ofcircuits that generate entangled systems of qubits are described above.For instance, Bell state generator 700, Type I fusion circuit 800 andType II fusion circuit 900 are examples of circuits that can generateentangled systems of qubits. Additional examples are described in WO2020/257772, “Photonic Computer Architecture.” Each entanglement circuit1902 can provide an input qubit to an N×2R raster mux circuit 1910. Forexample, qubits can be represented using a dual-rail encoding. Toprovide a qubit, an instance of entanglement circuit 1902 can have apair of output waveguides (corresponding to two rails that encode onequbit as described above) coupled to a pair of input waveguides ofraster mux circuit 1910. It should be understood that in FIGS. 19A and19B, a single coupling path (line) between circuit components representsa qubit. In embodiments using a dual-rail encoding, each coupling pathcan be implemented using a pair of waveguides. In embodiments usingother photonic encoding schemes, a coupling path can correspond to anumber of waveguides sufficient to encode one qubit. For example, in apolarization encoding, one waveguide may suffice to encode a qubit.

N×2R raster mux circuit 1910 can be similar to raster mux circuit 1200or other raster mux circuits described herein, except that each inputpath and each output path represents a qubit and may be implementedusing multiple waveguides. For instance, in a dual-rail encoding, rastermux circuit 1910 can include two identical N×2R GMZIs, one for each railof the qubit. Both GMZIs can be controlled by the same logic so thatboth rails of the same qubit propagate through raster mux circuit 1910.

In operation, for each time bin, control logic of raster mux circuit1910 can select the output path of one of the N entanglement circuits1902 as the active input path and can select one of the 2R output pathsas an active output path. Selection of the active input path can bebased on heralding signals received from each entanglement circuit 1902indicating whether that entanglement circuit 1902 successfully producedan entangled state. In some embodiments, there may be only one instanceof entanglement circuit 1902 (i.e., N can be equal to 1), in which casethe control logic of raster mux circuit 1910 may not need to select anactive input path. As with other raster mux circuits described herein,raster mux circuit 1910 can cycle through the R output paths 1936 duringa rastering period of 2R successive time bins such that raster muxcircuit 1910 can output a qubit onto output path 1936-1 during a firstcycle, output path 1936-2 during a second time bin, and so on until aqubit is output onto output path 1936-2R during the 2Rth time bin. Asindicated in FIG. 19A, qubits on output paths 1936-1 through 1936-R canbe interpreted as instances of “Qubit A” while qubits on output paths1936-(R+1) through 1936-2R can be interpreted as instances of “Qubit B.”It should be understood that qubits (photons) on different output paths1936 reach point 1920 at different times in a predictable, repeatablepattern: if a qubit on output path 1936-1 arrives at time t₁, then aqubit on output path 1936-k arrives at time t₁+kt_(c), where t_(c) isthe interval between time bins, as suggested by black dots 1906.

Turning to FIG. 19B, circuit 1900 also includes circuitry to performmeasurement operations on instances of Qubit A and instances of Qubit B.In this example, circuit 1900 includes a number of type II fusioncircuits (T2) 1952, an “X” measurement circuit 1954, and a “Z”measurement circuit 1956. Each type II fusion circuit 1952 can beconfigured to receive two qubits as inputs and perform a two-qubitmeasurement operation that consumes both input qubits, e.g., asdescribed above with reference to FIGS. 9A and 9B. As noted above, theinput qubits to type II fusion circuits 1952 are presumed to beentangled with other qubits (e.g., via operation of entanglementcircuits 1902 of FIG. 19A), and one effect of a successful type IIfusion operation is to “fuse” the respective systems of qubits withwhich the two input qubits are entangled into a single (larger)entangled system. Another effect of a type II fusion operation can bethe extraction of (classical) measurement data from the two-qubitmeasurement operation. X measurement circuit 1954 can perform asingle-qubit measurement in the Pauli X basis, and Z measurement circuit1956 can perform a single-qubit measurement in the Pauli Z basis.

Circuit 1900 also includes two GMZI circuits 1960, 1962. GMZI circuit1960 has R input paths 1959 coupled to receive the R instances of QubitA from raster mux circuit 1910 and 2R output paths 1961. One of theoutput paths 1961 of GMZI circuit 1960 is coupled to the input of Xmeasurement circuit 1954. The remaining 2R−1 output paths 1961 arecoupled to a set of delay lines 1964, each of which adds a differentamount of delay, from 0 to 2(R−1) time bins. The output of each delayline 1964 is coupled to a first input of one of type II fusion circuits1952. The number of instances of type II fusion circuit 1952 can beequal to the number of delay lines 1964, and in this example, there are2R−1 instances of type II fusion circuit 1952. GMZI circuit 1962 has Rinput paths 1963 coupled to receive the R instances of Qubit B fromraster mux circuit 1910 and 2R output paths 1965. One output path 1965is coupled to the input of Z measurement circuit 1956 The remaining 2R−1output paths 1965 are each coupled to a second input of one of type IIfusion circuit 1952. (As noted above, each path can be implemented usingone or more waveguides, depending on the particular qubit encoding.Where multiple waveguides are used to encode a qubit, each GMZI circuit1960, 1962 can be implemented using multiple identically configuredcopies of the same GMZI.)

Control logic 1970 can be implemented as a digital logic circuit with anarrangement of classical logic gates (AND, OR, NOR, XOR, NAND, NOT,etc.), such as a field programmable gate array (FPGA) orsystem-on-a-chip (SOC) having a programmable processor and memory, or anon-chip hard-wired circuit, such as an application specific integratedcircuit (ASIC). In some embodiments, an off-chip computer can be used toimplement control logic 1970, and in some embodiments, the same hardwarecomponents (including on-chip and/or off-chip components) can implementcontrol logic 1970 as well as the control logic for raster mux circuit1910.

In operation, for each time bin, control logic 1970 can select one ofthe input paths 1959 of GMZI 1960 as an active input path and can selectone of the output paths 1961 of GMZI 1960 as an active output path.Similarly, control logic 1970 can select one of the input paths 1963 ofGMZI 1962 as an active input path and can select one of the output paths1965 of GMZI 1962 as an active output path. Based on the selection,control logic 1970 can send control signals to GMZIs 1960 and 1962 toset the state of active switches within GMZIs 1960 and 1962 to couplethe active input path to the active output path.

Selection of an input path for each of GMZIs 1960 and 1962 can be basedon timing rules. For instance, as suggested by the black dots, qubitsarrive at different inputs of GMZI 1960 (or GMZI 1962) in different timebins, and the selection of an active input path can be based on a cyclecounter (e.g., as described above with reference to control logic 1230).Selection of the active output path can be based on an input signalindicating a desired disposition of each qubit. In some embodiments, oneinstance of Qubit A within a group of R instances and one instance ofQubit B within a group of R instances may be treated as a pair, and thedisposition can be either a type II fusion operation on the pair or asingle-qubit measurement on each qubit of the pair. The input signal canspecify which instance of Qubit B should be paired with each instance ofQubit A and whether the pair should be subject to type II fusion or tosingle-qubit measurements. In some instances, operation of entanglementcircuits 1902 (in FIG. 19A) may be non-deterministic, meaning that adesired entangled state is produced with a probability less than 1.Accordingly, there may be time bins during which no instance ofentanglement circuit 1902 generates the desired entangled state. In someembodiments, the determination of qubit pairings and/or the dispositionof a particular pair can depend on whether a usable entangled state wasgenerated by at least one of entanglement circuits 1902 during a giventime bin.

Based on information encoded in the input signal, control logic 1970 canselect an output path for each qubit instance. For example, where agiven instance of Qubit A is to be subject to single-qubit measurement,control logic 1970 can set the active switches in GMZI 1960 to couplethat instance of Qubit A to X measurement circuit 1954, and where agiven instance of Qubit B is to be subject to single-qubit measurement,control logic 1970 can set the active switches in GMZI 1962 to couplethat instance of Qubit B to Z measurement circuit 1956. Where aninstance of Qubit A and an instance of Qubit B are to be subject to typeII fusion measurement, those two qubits should arrive at the inputs ofthe same instance of type II fusion circuit 1952 simultaneously.However, due to the operation of raster mux circuit 1910, and due tovariability in which instance of Qubit A is paired with which instanceof Qubit B, paired instances of Qubit A and Qubit B may arrive at GMZIs1960 and 1962 at different times. Accordingly, control logic 1970 candetermine the number of time bins of delay to apply to the instance ofQubit A to allow the paired instance of Qubit B (which may be in a latertime bin as shown in FIG. 19A) to catch up. Control logic 1970 canselect the output path 1961 that couples to the appropriate delay line1964, and this selection also determines which instance of type IIfusion circuit 1952 will perform the fusion operation. Accordingly,control logic 1970 can select the output path 1965 for GMZI 1962 thatdelivers the instance of qubit B to the same instance of type II fusioncircuit 1952 that will receive Qubit A. As with control logic 1230described above, a lookup table can be provided such that, given aspecific pairing of one instance of Qubit A and one instance of Qubit Band a desired disposition for the pair (e.g., fusion or single-qubitmeasurements), the appropriate output paths (and corresponding activeswitch settings) for GMZIs 1960 and 1962 can be determined by a lookupoperation.

FIG. 20 is a spacetime diagram further illustrating the operation ofcircuit 1900 according to some embodiments. In this example, R=5. Shownat 2002 are the prescribed dispositions for each qubit instance: “X”denotes single-qubit X measurement; “Z” denotes single-qubit Zmeasurement; “T2” denotes type II fusion with a “priority” label definedsuch that the inputs to a single type II fusion operation are theinstance of Qubit A and the instance of Qubit B having the same prioritynumber. Shown at 2004 is a spacetime distribution of the qubits afteroperation of raster mux circuit 1910. The qubits are distributed inspace (on different paths) and in time. As shown at 2006, GMZI 1960applies delay to the instances of Qubit A that are designated for fusionoperations to bring them into temporal alignment with the pairedinstances of Qubit B. GMZI 1960 also routes instances of Qubit A thatare designated for single-qubit X measurement to X measurement circuit1954. As shown at 2008, GMZI 1962 provides spatial alignment ofinstances of Qubit B that are designated for fusion operations with thepaired instances of Qubit A. GMZI 1962 also routes instances of Qubit Bthat are designated for single-qubit Z measurement to Z measurementcircuit 1956. As shown at 2010, with the paired qubits in spatiotemporalalignment, type II fusion circuits 1952 can perform the fusionoperations.

It will be appreciated that circuit 1900 is illustrative and thatvariations and modifications are possible. A raster mux circuit canprovide any number R (2 or more) of outputs on different time bins. Insome embodiments, a time bin can be defined based on the speed at whichthe various circuit components can be operated. For instance, a detectormay incur deadtime after detecting a photon and the duration of a timebin can be selected to allow for detector deadtime. As another example,active optical switches (such as the switches in a GMZI) may have amaximum switching speed, and the duration of a time bin can be selectedso as not to exceed the maximum switching speed of the GMZIs. In someembodiments, after completing a raster period, an idle time may beintroduced to allow circuit components (e.g., detectors and/or photonsources) to recover.

In the example shown above, circuit 1900 includes 2R−1 delay lines 1964,which is sufficient to allow any instance of Qubit A to be paired withany instance of Qubit B. In some embodiments, fewer than 2R−1 delaylines can be used. Where this is the case, some pairings of instances ofQubit A and Qubit B might not be supported. For example if the time binis chosen to be shorter than the time needed to change the states of theactive switches in GMZIs 1960 and 1962, qubits may be provided at a ratefaster than the GMZIs can switch their routing. If the inputs for twofusion operations are too close in time, the desired routing may not beachievable. However, for some implementations, the density of fusionmeasurements may be low (e.g., where the success probability ofentanglement circuit 1902 is low), and the likelihood that fusionoperations would occur close in time may be negligible. More generally,to the extent that inability to support fusion operations betweencertain pairings of qubits is tolerable in a given system, the number ofdelay lines (and the number of fusion circuits) can be reduced, andGMZIs 1960, 1962 can be correspondingly reduced in size.

5. Generalized Mach-Zehnder Interferometer (GMZI) Implementations

In some embodiments, fast and low-loss optical switch networks canenable scalable quantum information processing using photonic qubits.More specifically, such networks can be employed within a linear-opticalquantum computing (LOQC) system, since many such systems relies onnon-deterministic processes of single-photon generation, entanglementgeneration and fusion measurements, and they also have importantapplications for quantum communications, such as enabling all-photonicquantum repeaters.

Advantageously, one or more embodiments disclosed herein provide for lowloss, fast, and minimally-decohering photonic switch networks. Someembodiments provide for switch networks having a minimization of depthand count and are particularly suited for implementations that includeactive phase shifters, which are historically the largest contributorsto the size and amount of noise in switch networks. Examples of switchnetworks will now be described. Such networks can be used, for instance,in any of the embodiments described above.

Components that can be used in photonic platforms include waveguides,directional couplers, passive and active (fast) phase shifters,crossings, single-photon detectors and heralded single-photon sources(HSPSs). S witch networks can be categorized according to their primaryfunction as follows. N-to-1 (M) muxes (also referred to as N×1 muxes)map one (or multiple M) inputs to designated output ports. The inputsare commonly assumed to be probabilistic and of the same type, althoughmore complicated assumptions apply in some problems. For example, aN-to-4 photon mux extracts groups of four photons from N HSPSs.Sometimes it is necessary to carefully distinguish the number of output(input) ports from the number of principal target outputs (inputs). Mostcommonly, the excess ports must be populated with the vacuum state, andthe switch network is required to access specific distributions(“patterns”) of the outputs (inputs) across the ports. We refer toswitch networks as permutation networks when their primary purpose is torearrange (subsets of) inputs, where the inputs should generally beregarded as inequivalent. Furthermore, switch networks are alsoclassified on the basis of the photonic degree of freedom distinguishingtheir inputs. Schemes based on space and time are the most common, butthe use of frequency, orbital angular momentum, and combinations ofmultiple degrees of freedom has also been proposed.

In some embodiments, Mach-Zehnder Interferometers (MZIs) may be usedwhich are networks that implement identity or swap operations on twoinputs. Two possible realizations of this type of circuit are shown inFIGS. 21A and 21B. FIGS. 21A and 21B show building blocks of compositeswitch networks. FIGS. 21A and 21B show 2-to-2 MZIs that implementidentity or swap operations on the inputs. The circuits consist of twodirectional couplers with an active phase shifter (gray) on one or botharms between them. The push-pull configuration shown in FIG. 21A alsohas a fixed passive −π/2 phase shift (white) on one arm and selectsbetween the two operations by setting the top or bottom active phase to−π/2. The configuration shown in FIG. 21B uses a 0 or −π active phase toselect the operation. Many switch network architectures are built byconnecting multiple MZIs to form various topologies.

The Generalized Mach-Zehnder Interferometer (GMZI) is an extension of anMZI with N>2 inputs and M≥1 outputs, shown in FIG. 21C. Thisconfiguration allows a set of permutations to be performed on theinputs, as discussed in further detail below, making this device apowerful block for the construction of composite N-to-1 and N-to-Mswitch networks. FIG. 21C shows a N-to-M GMZI made of two passivebalanced splitter networks (white) and a layer of N active phaseshifters (gray). Varying the settings of the active phases selectsspecific permutations of the N inputs and routes them to M>1 outputports.

There are a number of spatial mux schemes that select one of multipleinputs from distinct locations in space. For example, a N-to-1 GMZI canbe used as a mux, since it allows routing of any input to a singleoutput port. The advantages of this scheme are its low constant activephase shifter depth (1) and count (N). However, the total propagationdistance and the number of waveguide crossings increase rapidly with N.This downside of the monolithic GMZI structure is obviated byconstructing composite switch networks of 2-to-1 MZIs, at the cost ofincreasing the component depth and count. Two examples of N-to-1 schemesof this kind include the “log-tree” and “chain”, both of which can bebuilt with no crossings.

FIGS. 22A and 22B show spatial N-to-1 muxes, with inputs at Nspatially-distinct locations (ports). FIG. 22A shows a log-tree mux (N=8example). 2-to-1 MZIs form a tree structure with 2(2^(┌log) ² ^((N)┐)−1)active phase shifters arranged in ┌log₂ (N)┐ layers. FIG. 22B shows achain mux (N=4 example). (N−1) MZIs are connected through one output andinput to form a line. The active phase shifter count is the same as forthe log-tree, but the depth varies between 1 and (N−1).

In a “log-tree”, the MZIs form a converging symmetric tree of degree 2,where the chosen input is routed from one of the leaves to the root, asshown in FIG. 22A. An asymmetric variant of this scheme, known as a“chain”, includes MZIs cascaded to form a linear topology in which eachblock selects either the output of the previous block or the new input,as shown in FIG. 22B. The depth of the network traversed by the outputdepends on the chosen input, which can worsen the interference ofresources from different chains, due to imbalanced losses and errors.The switching logic of this scheme presents an interesting advantage:while being very simple and entirely local to each individual MZI, itminimizes the amount of error on by selecting the input availableclosest to the output. Analysis of these three schemes in the context ofsingle photon multiplexing shows that all three architectures requirecomponents with performance well beyond the state-of-the-art to achievea multiplexing efficiency high enough for use in LOQC.

In temporal multiplexing, resources can be input at the same spatiallocation but different times, and the aim is to produce an output in aspecific time bin. This requires networks with fewer components, but theoutput time bins become longer. There are two main kinds of temporalschemes: designs with storage devices, such as cavities or fiber loops,and designs based on networks of delays The former simply consist of astorage device and a single 2×2 switch network used to choose whether tostore or output each input, as shown in FIG. 23A. This can be thought ofas the temporal version of a chain mux, and it presents the sameadvantage in terms of switching logic. The log-tree also has a temporalequivalent known as a “binary-division delay network”. This schemeconsists of a series of MZIs with delays of different lengths betweenthem, as illustrated in FIG. 23B.

FIGS. 23A and 23B show N-to-1 temporal muxes, with inputs in N distincttime bins. FIG. 23A shows a storage loop scheme (time chain). A 2×2 MZIreceives one resource per time bin T and routes it to a storage device(a delay line here) or discards it. After N time bins, the chosen inputis output. The number of active phase shifters in the path of the choseninput varies between 1 and N. FIG. 23B shows a binary delay network(time log-tree). The scheme comprises a series of ┌log₂ (N)┐+1 MZIs withdelays of lengths 2^(n)T between them, where T is the duration of a timebin at the input and n=0, . . . ┌log₂ (N)┐−1. The active phase shifterdepth scales as with the number of input time bins as ┌log₂ (N)┐.

The topologies described above can be generalized by replacing each MZIwith a GMZI with n inputs, as shown in FIGS. 24A-24D. This introduces atrade-off between the active phase shifter depth and count, whichdecreases with n, and the number of waveguide crossings and propagationdistance within each block, which increases with n. In addition, thismodification turns temporal schemes into hybrid networks, where multiplespatially distinct resources are input in each time bin. The trade-offsintroduced by the parameter n can be exploited to optimize the structureof these schemes for different regimes of physical error rates.

FIGS. 24A-24D show examples of generalized N-to-1 composite multiplexingnetworks, obtained by replacing the MZI sub-blocks with n×1 GMZIs. FIG.24A shows a generalized spatial log-tree (n=3 example with some firstlayer GMZIs omitted for simplicity). The degree of the tree is n and itsdepth is ┌log_(n) N┐. FIG. 24B shows a generalized spatial chain. Eachstage after the first takes n−1 new inputs, so that the depth of thenetwork varies between 1 and ┌(N−1)/(n−1)┐. FIG. 24C shows a generalizeddelay network (time log-tree). The GMZIs enclose ┌log_(n) N┐ layers ofn−1 delays with lengths n^(i), . . . (n−1)n^(i), where i=0, . . . ,┌log_(n) N┐−1 is the index of the layer of delays. The number of activephase shifters on a path across the scheme is ┌log_(n) N┐+1. FIG. 24Dshows a generalized storage loop scheme. n−1 inputs enter the GMZI inevery time bin. After ┌N/(n−1)┐ time bins, the GMZI outputs the choseninput.

In applications such as LOQC, which rely on the interference ofmultiplexed resources, multiplexing is used to produce synchronizedoutputs. The schemes described so far achieve this by having a singlepredetermined output spatio-temporal bin. However, when large outputprobabilities are needed this leads to a large of resources, which canbe understood as follows. The number of available resources for anetwork of size N follows a binomial distribution with average valueN=Np, where p is the probability of an input being populated. Theprobability of a network successfully producing an output is thenp_(mux)=1−(1−p)^(N). For the typical situation with large N and small pvalues, the binomial distribution is well approximated by a Poissoniandistribution, and so p_(mux)≃1−e^(−Np). It follows that the averagenumber of inputs scales as Np=−ln(1−p_(mux)), and so the number ofavailable resources that are not used grows rapidly as p_(mux)approaches 1. An alternative approach that leads to major efficiencyimprovements is relative multiplexing. Rather than routing resources tosingle pre-allocated outputs, this technique uses spatial or temporallog-tree networks to synchronize selected inputs in variable space-timelocations, chosen depending on the resources available at any particularinstant.

N-to-M schemes in the literature are generally based on the spatialdegree of freedom. The simplest of these is a GMZI with more than oneoutput, which has the appealing feature of a single layer of N activephase shifters. However, it only gives access to N permutations, andtherefore to limited combinations of inputs. Consequently, the N×M GMZIis more useful when used as a permutation network or as a building blockfor larger schemes. More flexible routing is achieved by using smallernetworks to build composite topologies, known as “switch fabrics”.However, the component depth and count and the size of the crossingnetworks of these schemes tend to be large, and these downsides tradeagainst each other, making the networks impractical for use in the fieldof quantum applications.

As an example, Spanke's tree network, shown in FIG. 25A, allowsarbitrary rerouting of the inputs with a constant active switch depth of2, at the cost of a large number of active phase shifters and waveguidecrossings. However, the number of active phase shifters and waveguidecrossings scales as O (NM). On the other hand, the scheme shown in FIG.25B avoids large crossing networks, but has an active phase shiftercount O (NM) and depth that varies between 1 and M, resulting invariable error rates on the outputs.

FIGS. 25A and 25B show examples of N-to-M switch networks. FIG. 25Ashows a Spanke network. Two layers of interconnected GMZIs allowarbitrary routing of N inputs to M outputs. The fixed active phaseshifter depth of 2 makes this scheme interesting, but the scaling of thenumber of active phase shifters and crossings scaling as (NM) poseschallenges for large sizes. FIG. 25B shows a concatenated GMZI. Thisscheme consists of M concatenated GMZIs with progressively feweroutputs. No complex crossing networks are required between its buildingblocks, but the O(NM) active phase shifter count and variable depth upto M limit the maximum feasible network size.

For quantum applications, where low error rates are required, N-to-Mmuxes need to be simplified to reduce the number of active phaseshifters, both in total and along the path to the output, as well as thecomplexity of the crossing networks. The routing algorithms associatedwith these networks also need to be simplified, to avoid the need forunfeasibly long delays for the inputs. The complexity of the logic islargely determined by its generality, so restricting the operation ofthe networks to specific tasks is helpful to reduce processing times.These provide guiding principles for the design of additional schemes.

A general switch network implements a set of unitary transfer matricesU_(k), where each unitary routes light between a subset of input andoutput ports. If U_(k) routes light from port t to port s, then its sthrow and tth column must be zero apart from |U_(s,t)|=1, and similarlyfor other pairings of input and output ports. The aim of this section isto elucidate the sets of routing operations that are achievable usingthe simplest form of a many-mode switching network, which is to say onecorresponding to transfer matrices U_(k)=WD_(k)V^(†), where the unitarymatrices W, V^(†) describe passive interferometers, and the D_(k) form aset of diagonal phase matrices. The phase matrices are implementedphysically using a single layer of fast phase shifters acting on everymode, and for simplicity, we will write D in terms of a phase vector d,D_(s,t)=d_(s)δ_(s,t). The discussion below provides a comprehensivetreatment of these switch networks and presents several newconstructions.

An important class of switch networks is obtained by considering sets ofpermutation matrices {U_(k)=WD_(k)V^(†)}. By adding the fixed passivenetwork corresponding to e.g. U₁ ⁻¹ (so, the inverse of an arbitrarypermutation from that set), we obtain a new set {U_(k)U₁⁻¹}={WD′_(k)W^(†)} of pairwise commuting permutation matrices. So itmakes sense to restrict the discussion to the case where the {U_(k)} arecommuting. Switch networks of this type were introduced above as“generalized Mach-Zehnder interferometers” (GMZIs). Here we need a moreprecise definition for GMZIs, and we will define them as switch networkshaving the following specific properties:

-   -   (i) {U_(k)=WD_(k)W^(†)} is a set of transfer matrices        corresponding to commuting permutations of N modes. The entries        of D_(k) are given by roots of unity (up to an overall global        phase factor e^(iϕk) which can be chosen at will).    -   (ii) The GMZI switch setting D_(k) routes light from input port        1 to output port k.

From these properties it is straightforward to prove that the GMZI musthave exactly N settings, and that for any choice of input and outputport, there is exactly one setting which routes light between the ports.

From a mathematical standpoint, the set of operations implemented by aGMZI on N modes forms an abelian group of order N. This fact is veryhelpful here as it allows us to characterize the entire family of GMZIsdefined by (i), (ii) using well-known results from group theory (namelythe basis theorem for finite abelian groups). In particular, for anyGMZI, {U_(k)} must be isomorphic to a direct sum of cyclic groups, wherethe order of each of the cyclic groups is a power of a prime number.

To be more concrete, we define groups of commuting permutations

[n₁, n₂, . . . , n_(r)]) generated by matrices C^((n) ¹ ⁾⊗I^((n) ²⁾⊗I^((n) ³ ⁾ . . . , I^((n) ¹ ⁾⊗C^((n) ² ⁾⊗I^((n) ³ ⁾ . . . , I^((n) ¹⁾⊗I^((n) ² ⁾⊗C^((n) ³ ⁾ . . . , where (C^((n)))_(i,j)=δ_(i,(j+1 mod n))is a cyclic permutation matrix of size n, and I^((n) ^(l) ⁾ is then_(l)×n_(l) identity matrix, and ⊗ is the Kronecker product on matrices(The Kronecker product here acts at the level of linear-optical transfermatrices and should not be confused with tensor product operations onquantum state spaces), and the group operation is matrix multiplication.Then, any GMZI on N modes, satisfying properties (i), (ii) above, mustimplement a set of permutation operations which corresponds to one ofthe possibilities for

([n₁, n₂, . . . , n_(r)]) with N=Π_(l=1) ^(r)n_(l) (up to fixed modepermutations at the input and output).

The different types of GMZIs of fixed size can now be determined usingthe fact that

([n₁, n₂]) and

([n₁n₂]) are isomorphic if and only if n₁ and n₂ are coprime. Forexample, for N=8, we can identify three fundamentally different types ofGMZI:

-   -   (i)        ([2,2,2]), permutations are generated by Pauli matrices        X⊗I⁽²⁾⊗I⁽²⁾, I⁽²⁾⊗X⊗I⁽²⁾, I⁽²⁾⊗I⁽²⁾⊗X.    -   (ii) {        ([4,2])}, permutations are generated by matrices

${{{C^{(4)} \otimes I^{(2)}}{where}C^{(4)}} = \begin{pmatrix} & & & 1 \\1 & & & \\ & 1 & & \\ & & 1 & \end{pmatrix}},{{and}{I^{(4)} \otimes {X.}}}$

-   -   (iii)        ([8]), permutations are generated by matrix

$c^{(8)} = {\begin{pmatrix} & & & & & & & 1 \\1 & & & & & & & \\ & 1 & & & & & & \\ & & 1 & & & & & \\ & & & 1 & & & & \\ & & & & 1 & & & \\ & & & & & 1 & & \\ & & & & & & 1 & \end{pmatrix}.}$

We refer to GMZIs implementing

([2,2, . . . ,2]), i.e. permutations of the form of swaps on subsets ofmodes, as “Hadamard-type” GMZIs due the type of passive interferometerwhich is used (explained below). Similarly, we refer to GMZIsimplementing

([N]) as “discrete-Fourier-transform (DFT)-type”.

The discussion above characterizes the routing power of linear-opticalcircuits using one-layer of fast phase shifters in the switch network.In particular, a GMZI on N modes is limited to N routing operations,which is obviously small compared to the N! possible mode rearrangementoperations. However, the possibility of implementing different sets ofpermutation operations is exploited by some of designs for spatial andtemporal muxes which are discussed herein. Strictly speaking thelimitation to N operations originates in property (ii) above—i.e. theability to route light from any input port to any output port. Moregeneral constructions using a single stage of active phase shifts can betrivially obtained by acting with separate GMZIs on subsets of modes.The resulting transfer matrices are given by the direct sum of theindividual GMZIs' transfer matrices. For example, using three MZIs inparallel results in a switch network on 6 modes, allowing 8 differentsettings. Such a construction can implement abelian groups ofpermutations of maximum order, which are given in J. M. Burns and B.Goldsmith, Bull. London Math. Soc. 21, 70 (1989), with the number ofoperations scaling to good approximation as ˜3^(N/3).

We now turn to linear-optical circuits that can implement the GMZIsdefined above. In particular, a circuit that can implement the routingoperations

([n₁, n₂, . . . , n_(r)]) on N=Π_(l=1) ^(r)n_(l) modes must enacttransfer matrices of the form,

P _(k)=(C ^((n) ¹ ⁾)^(k) ¹ ⊗(C ^((n) ² ⁾)^(k) ² ⊗ . . . ⊗(C ^((n) ^(r)⁾)^(k) ^(r) ,

with settings vector k where 0≤k_(l)<n_(l) with l=1, . . . , r. This canbe achieved using a circuit with transfer matrices WD_(k)W^(†) asfollows:

W = W^((n₁)) ⊗ W^((n₂)) ⊗ … ⊗ W^((n_(r)))${{{with}\left( W^{(n_{l})} \right)_{s,t}} = \frac{e^{l2\pi{st}/n_{l}}}{\sqrt{n_{l}}}},$

where the W^((n) ^(l) ⁾ are DFT matrices; the kth setting of the fastphase shifters is given by

D _(k) =D _(k) ₁ ^((n) ¹ ⁾ ⊗D _(k) ₂ ^((n) ² ⁾ ⊗ . . . ⊗D _(k) _(r)^((n) ^(r) ⁾,

with(d _(k) ^((n)))_(s) =e ^(−l2πks/n)for D _(k) ^((n)).

One route to constructing practical interferometers for W and W^(†) isto reduce them to networks of beam-splitter and phase-shifter componentsusing generic unitary decompositions from M. Reck et al., Phys. Ref.Lett. 73, 58 (1994), or W. R. Clements et al., Optica 3, 1460 (2016).These decompositions have optical depth (number of optical elementsencountered on the longest path through the interferometer) scaling as2N−3 and N respectively. This means that the transmittance along thelongest path will scale with an exponent which is proportional to thesize parameter N—which presents a severe experimental limitation forscaling to large GMZI sizes.

GMZI networks—having a lot of special structure—allow for specificdecompositions of the type given by equation 2600 shown in FIG. 26 ,where the matrices S correspond to crossing networks which reorder modeswithin the interferometer. Since the subexpressions of the form I^((N/n)^(l) ⁾⊗V^((n) ^(l) ⁾ correspond to repeated blocks of modes interferingaccording to unitary V^((n) ^(l) ⁾, the equation for Win FIG. 26 can beseen to describe stages of local interference separated by crossingnetworks. Note also that since the bracketed expressions in thedecomposition commute there is some freedom in the configuration of thecrossing networks, and some of them can be treated as relabelings ofmodes rather than physical circuit elements. FIG. 27A illustrates theconstruction of a Hadamard-type GMZI using the decomposition, as well assimplification which is possible when the GMZI is used as a N-to-1 mux.

FIGS. 27A and 27B show Hadamard-type GMZI constructions: (i) in FIG.27A, illustration of a linear-optical circuit for a GMZI on N=16 modes,for which the fast phase shifters are set to configurations of 0 and πto select one of 16 operations from

([2,2,2,2]); (ii) in FIG. 27B, possible simplification of the circuitwhen only one output port is required— as is the case when the GMZI isused as a N-to-1 mux. The passive interferometers are constructedfollowing the decomposition of W with stages of interference using 50:50beam-splitters or directional couplers on pairs of adjacent modes,separated by crossings networks. Note that the phases in the physicalinterferometer generally differ from the constructions given in the maintext, and this implies minor modifications for the transfer matrices andphase-shifter settings.

For more general GMZI types, we note that the unitary matrices V^((n)^(l) ⁾ can be decomposed into elementary beam-splitter and phase-shifteroperations using the generic decomposition methods mentioned above.Alternatively, since the V^((n) ^(l) ⁾ are assumed to be discreteFourier transforms, they can be recursively decomposed into smallerdiscrete Fourier transforms acting on sets of local modes I^(n) ^(l)^(/(n) ^(l) ^(′))⊗V^((n) ^(l) ^(′)), I^(n) ^(l) ^(/(n) ^(l) ^(″))⊗V^((n)^(l) ^(″)) (for any sizes satisfying n_(l)=n_(l)′×n_(l)″) together withcrossings networks and additional phase shifts.

One more subtle feature of the GMZI constructions that was remarked onabove is that the matrices D_(k) for the GMZIs are determined up to asetting-dependent global phase factor e^(iϕk). In principle these globalphases can be freely set over a range [0,2π) (provided the active phaseshifters themselves are configured with sufficient phase range). For anapplication such as single-photon multiplexing, the global phase factorshave no role in the operation of the switch network. However, they canbe useful if the switch network is applied to only some part of theinput states (e.g. single rails from dual-rail qubits) or if it isincorporated in larger interferometers. In these cases, additionalfunctionality can be absorbed into the operation of the switch networkwithout adding extra layers of switching.

This idea is very useful for LOQC, where it is often desirable tomultiplex some circuit which generates entangled states, whilst alsoapplying internal adaptive corrections to its output. An example of thisoccurs when multiplexing Bell states from a standard BSG circuit. Thiscircuit produces a Bell state across four modes with probability 3/16,but the Bell states do not conform to dual-rail qubit encoding (i.e.with qubits allocated to fixed pairs of modes) in a third of cases.Although this problem can be addressed using an additional MZI at themux output to perform an optional mode-swap operation, a more elegantsolution is presented in FIGS. 28A and 28B.

FIGS. 28A and 28B show examples of larger GMZI to implement adaptiveswaps of rails while multiplexing Bell states generated with n₂ standardBSGs. FIG. 28A shows sending the two rails that might need to be swapped(circled in red) through a single GMZI of size N=n₁n₂ (n₁=n₂=2 in thisdiagram) allows multiplexing and permutation operations to be combinedwhile avoiding the need for an additional switching stage. FIG. 28Bshows that the modular structure of the GMZI can be exploited to applyportions of the circuit at different locations and to optimize thephysical implementation. In this example, the network which incorporatesthe swap operation can be decomposed into two 2-to-1 GMZIs with extradirectional couplers applied at the output of the BSGs and between thetwo output rails.

In this approach, a mux on n 2 copies of the BSG implements multiplexingand swap operations, using a size N=n₁n₂ GMZI on n 1=2 inner rails fromeach BSG, and regular n₂-to-1 multiplexing for the outer rails. Theability to permute the rails increases the success probability forgenerating a dual-rail encoded Bell state from ⅛ to 3/16, and therebydecreases the amount of multiplexing needed to reach any particulartarget output probability by a factor of ˜1.55.

More generally, the transfer matrices associated with a GMZI thatimplements the routing operations

([n₁, n₂]) are

$\begin{matrix}{P_{({k_{1},k_{2}})} = {\left( C^{(n_{1})} \right)^{k_{1}} \otimes \left( C^{(n_{2})} \right)^{k_{2}}}} \\{= {\left( {C^{(n_{1})} \otimes I^{(n_{2})}} \right)^{k_{1}}\left( {I^{(n_{1})} \otimes C^{(n_{2})}} \right)^{k_{2}}}}\end{matrix}.$

This can be interpreted as n 1 separate copies of n₂-to-1 GMZIs (secondterm) with an additional set of permutations of the n₁ outputs alsoavailable (first term). So, permutations of n₁ rails can be implementedwhile multiplexing each one n₂ times by sending all N=n₁n₂ inputsthrough a single larger GMZI rather than smaller separate ones. The keyadvantage of this method is that the depth and total number of activephase shifters do not change (1 and N respectively).

Using a larger GMZI comes at the cost of increasing the optical depth ofthe circuit, particularly in terms of waveguide crossings. As seen fromthe expression of W above, the passive interferometers in a GMZI can bedecomposed into smaller networks connected by layers of crossings. Thismodular structure can be exploited to distribute parts of the circuitacross different locations and avoid large on-chip crossing networks. Inthe BSG example, the implementation shown in FIG. 28B highlights how thefirst layer of crossings can be realized in a different way, e.g. usinglong distance phase-stable optical routing, to mitigate the impact ofthe largest crossing network in the interferometer.

The discussion so far presented a large family of GMZIs and explainedtheir key properties, taking an approach focused on achievable sets ofpermutations which is different to earlier works. As well as N-to-1muxing (potentially with extra functionality as explained above, theseGMZIs have assorted applications as building blocks for spatial andtemporal muxes. Alternative constructions of GMZIs are also possible,and it is valuable to explore them with a view to minimizing practicalrequirements on fast phase shifters. However, it is not feasible toexhaust all possible GMZI designs, as some properties for Hadamardmatrices are not known. Instead we will highlight some specific newconstructions with useful properties.

One observation is that phase swing requirements (where the swing isdefined per phase shifter as the difference between the maximum andminimum phase shifts across all GMZI settings) can sometimes be reducedby introducing fixed phase-shift offsets. For some of the constructionsabove, the phase shifter settings correspond to complete sets of rootsof unity, and the phase swing is π for Hadamard interferometers and >πfor the other GMZI types. Table 1 shows examples of reduced swing forGMZI sizes N=2,3,4 including examples of GMZIs with reduced phase swingusing fixed phase-shift offsets. It is assumed that all the fast phaseshifter components are identical and access the same range of phaseshifts (which is minimized). Note that the use of offsets necessitatesmodification of the GMZI transfer matrices by additional phasefactors—corresponding to setting-dependent “global” phases at theoutput.

TABLE 1 GMZI Phase type offsets Comment Hadamard (−3π/2, 0) Swingreduced from π to π/2, N = 2 coinciding with MZI variant in FIG. 21A.DFT N = 3 (−4π/ Swing reduced from 4π/3 to 2π/3. 3, 0, 0) Hadamard (−π,0, 0, 0) Swing unchanged at π, N = 4 but for each setting only one phaseshifter is set to π and the others to 0.

To find some more subtle constructions, we can consider generalconstraints on GMZIs implementing transfer matrices U_(k)=WD_(k)V^(†) onN modes, which are required to act minimally as N-to-1 muxes. It isstraightforward to prove a lemma stating that (a), V in this case mustbe proportional to a complex Hadamard matrix (i.e. V must satisfy|V_(s,t)|=1/√{square root over (N)} as well as being unitary), and (b)the phase vectors d_(k) must be orthogonal. A simple consequence of thisresult is that it is never possible to construct any GMZI for which thephase-shifter swing is less than π/2 (since it is never possible toachieve 0 for the real part of

d_(k), d_(k′)

). Similarly, when the phase-shifter values are restricted to {0,π/2} itis not possible to find more than 2 orthogonal vectors d_(k) for anyeven value of N (and never more than 1 for odd values of N), which is tosay that it is not possible to do better than a 2-to-1 mux.

As another application of this lemma, one can look for sets oforthonormal phase vectors {d_(k)} and construct a GMZI which uses theseas phase settings for a N-to-1 mux, by choosing V to have row vectorsv_(k)=d_(k), and any unitary W with first row vector w₁=(1,1, . . .,1)/√{square root over (N)}. An interesting and non-trivial example ofsuch a set of phase vectors is given in Table 2. More specifically theable below shows examples of six orthogonal phase vectors with a subsetd₁, . . . , d₄ having a reduced phase swing of 2π/3 (compared to 4π/3for the entire set). A N=6 GMZI constructed using these settings canimplement a 4-to-1 mux which has phase swing of only 2π/3 (byrestricting to the first four phase-shifter settings). Furthermore, itis easily seen that this example is not related to the constructionsabove since the only possibility would be the GMZI implementing

([6])≅

([3,2]), for which individual phase settings range on six values(compared to three in Table 2).

TABLE 2 Settings for a N = 6 GMZI acting as a 6-to-1 mux d₁ = (1, 1, 1,e^(−21π/3), e^(−21π/3), e^(−21π/3))/{square root over (6)} d₂ = (1,e^(−21π/3), e^(−21π/3), 1, e^(−21π/3), 1)/{square root over (6)} d₃ =(e^(−21π/3), 1, e^(−21π/3), e^(−21π/3), 1, 1)/{square root over (6)} d₄= (e^(−21π/3), e^(−21π/3), 1, 1, 1, e^(−21π/3))/ {square root over (6)}d₅ = (1, e^(−21π/3), e^(−41π/3), e^(−21π/3), 1, e^(−41π)/3) /{squareroot over (6)} d₆ = (e^(−21π/3), 1, e^(−41π/3), 1, e^(−21π/3),e^(−41π)/3)/{square root over (6)}

Finally, we turn to a new way of using GMZIs when phase settings aremodified from those connecting single input and output ports. TakingHadamard-type GMZIs with transfer matrices U_(k)=WD_(k)W^(†) on N modes,consider first when the phase vector d_(k′) for D_(k′) is modified sothat −π phases are set to a (common) value −ϕ, while the 0 phases areunchanged. In this case U_(k′) is modified to

${{\overset{\sim}{U}}_{k^{\prime}}(\phi)} = {{e^{{- i}\phi/2}\left\lbrack {{{\cos\left( \frac{\phi}{2} \right)}I^{(N)}} + {i{\sin\left( \frac{\phi}{2} \right)}U_{k^{\prime}}}} \right\rbrack}.}$

This unitary maps a single photon incident at one input port to asuperposition across the mode at the input and the output under thepermutation U_(k), with weighting controlled by the value of ϕ. Furthermodification of the phase settings can achieve mappings from one inputto arbitrary pairs of output ports—suppose it is desired to map frominput port p₁ to output ports q₁ and q₂, then this can be implemented byfinding the (unique) settings k₁, k₂ with U=WD_(k) ₁₍₂₎ W^(†):p

q₁₍₂₎, and choosing phase vector

$\overset{˜}{d} = {{e^{{- i}\phi/2}\left\lbrack {{{\cos\left( \frac{\phi}{2} \right)}d_{k}} + {i{\sin\left( \frac{\phi}{2} \right)}d_{k^{\prime}}}} \right\rbrack}.}$

The transfer matrix for the GMZI is then

${{\overset{\sim}{U}(\phi)} = {e^{- \frac{i\phi}{2}}\left\lbrack {{{\cos\left( \frac{\phi}{2} \right)}U_{k}} + {i{\sin\left( \frac{\phi}{2} \right)}U_{k^{\prime}}}} \right\rbrack}},$

where the individual phase settings are taken from the set {0, −ϕ, −π,−π−ϕ}. Note that a second input port p₂ is also mapped to the pair q₁and q₂, where U_(k)U_(k′):p₁

p₂. We call a GMZI used according to the equation above for Ũ(ϕ) aswitchable pairwise coupler and it can be useful in spatial and temporalmuxes (with the proviso that paired ports receive the vacuum state toavoid contamination of the intended input).

6. Additional Embodiments

The foregoing examples of raster mux circuits and their applications areillustrative and can be modified as desired. Although some examples maymake reference to use-cases related to quantum computing, where photonspropagating in waveguides may be used to encode qubits, it should beapparent from this disclosure that raster mux circuits are applicable inany photonic circuit where temporal and/or spatial rearrangement ofphotons is desired. Further, raster mux circuits can be used foraligning a group of photons on different paths into any targetspatiotemporal relationship, provided that an appropriate combination ofoutput paths (including delay lines where applicable) is provided. Thesize of a time bin, the number of spatial and/or temporal modes, and thenumber of photons can be varied as desired.

As noted above, in some embodiments, “errant” photons can occur. Forinstance, in a given time bin, a raster mux circuit may produce a secondphoton on an output path other than the intended output path. Varioustechniques can be used to address errant photons. For instance, blockingswitches as described above can be used to prevent errant photons frompropagating into downstream circuits; the blocking switches can be setto permit. As another example, clocked electrical gating can be used toignore signals from particular downstream detectors except during timebins when signals are expected from those detectors.

As described above, a raster multiplexer can include a set (alsoreferred to “raster group”) of output paths that are selected in arasterized manner such that each output path in the raster group isselected as an active output path once during a raster period. Theraster period can include a set of consecutive time bins. In otherembodiments, selection of an active output path can be based on a timingsignal such that different output paths in the raster group are selectedat different times (not necessarily on consecutive cycles). Theselection of an output path can be cyclic, such that the active outputpath is selected according to a fixed order, and independent of theselection of an active input path. In some embodiments, a rastermultiplexer can also include one or more other output paths in additionto the raster group. The control logic can have multiple operatingmodes. For example, in a “rastering” mode, the control logic can selectamong the raster group in a manner as described above. In a“non-rastering” mode, the control logic can implement other algorithmsto select an output path and may select from any output path includingoutput paths that are in the raster group and/or output paths that arenot in the raster group.

Further, embodiments described above include references to specificmaterials and structures (e.g., optical fibers), but other materials andstructures capable of producing, propagating, and operating on photonscan be substituted. Raster multiplexing is described above in thecontext of optical/photonic circuits; however similar techniques may beapplied to other types of propagating signals.

Control logic to control the switches and other optical componentsdescribed herein can be implemented as a digital logic circuit with anarrangement of logic gates (AND, OR, NOR, XOR, NAND, NOT, etc.), such asa field programmable gate array (FPGA) or system-on-a-chip (SOC) havinga programmable processor and memory, or an on-chip hard-wired circuit,such as an application specific integrated circuit (ASIC). Control logiccan be implemented on-chip with the waveguides, beam splitters,detectors and/or and other photonic circuit components or off-chip asdesired. In some embodiments, photon sources, raster mux circuits,and/or other optical circuits can be coupled to an off-chip computersystem having a processor and a memory, and the off-chip computer systemcan be programmed to execute some or all of the control logic.

It should be understood that all numerical values used herein are forpurposes of illustration and may be varied. In some instances ranges arespecified to provide a sense of scale, but numerical values outside adisclosed range are not precluded. Terms such as “synchronized” or“simultaneous” (or “same” or “identical”) should be understood in theengineering rather than the mathematical sense: finite design tolerancescan be defined, and events separated by less than the design tolerancemay be treated as synchronized or simultaneous. A “time bin” refers to atemporal mode that distinguishes different photonic states in the samewaveguide (or spatial mode). The duration of a time bin can be definedbased on characteristics of the optical circuits (e.g., there may besome variation in the delay between pumping a photon source andobtaining an output photon from the source), and successive time binscan be separated by arbitrary time periods (e.g., to allow circuitcomponents to recover or change state before receiving the next photon).

It should also be understood that all diagrams herein are intended asschematic. Unless specifically indicated otherwise, the drawings are notintended to imply any particular physical arrangement of the elementsshown therein, or that all elements shown are necessary. Those skilledin the art with access to this disclosure will understand that elementsshown in drawings or otherwise described in this disclosure can bemodified or omitted and that other elements not shown or described canbe added. The terms “upstream” and “downstream” as used herein refer tothe direction of photon propagation through an optical circuit (from“upstream” inputs toward “downstream” outputs) and may correspond to anydirection in physical space.

This disclosure provides a description of the claimed invention withreference to specific embodiments. Those skilled in the art with accessto this disclosure will appreciate that the embodiments are notexhaustive of the scope of the claimed invention, which extends to allvariations, modifications, and equivalents.

1. (canceled)
 2. A system comprising: a multiplexer circuit having: aplurality of input paths to receive photonic qubits; a plurality ofoutput paths including a first subset of output paths having a number(R) of output paths and a second subset of output paths having thenumber R of output paths, wherein R is at least 2; and an opticalswitching network coupled between the input paths and the output paths,the optical switching network comprising a plurality of active opticalswitches arranged to selectably couple a photonic qubit from any one ofthe input paths to any one of the output paths; a plurality ofdownstream circuits, each downstream circuit having a first input portand a second input port; a first optical switching circuit having aplurality of input paths coupled to a first subset of the output pathsof the multiplexer circuit and a plurality of output paths coupled tothe first input ports of the downstream circuits via delay circuits thatintroduce different amounts of delay; a second optical switching circuithaving a plurality of input paths coupled to a second subset of theoutput paths of the multiplexer circuit and a plurality of output pathscoupled to the second input ports of the downstream circuits; andcontrol logic coupled to the multiplexer circuit, the first opticalswitching circuit, and the second optical switching circuit andconfigured to control the multiplexer circuit, the first opticalswitching circuit, and the second optical switching circuit such that afirst photonic qubit output on one of the first subset of output pathsof the multiplexer circuit and a second photonic qubit output on one ofthe second subset of output paths of the multiplexer circuit arrive intemporal alignment at the first and second input ports of one of thedownstream circuits.
 3. The system of claim 2 wherein the control logicis further configured to: receive, for a time bin of a plurality of timebins, an input signal indicative of whether a photonic state encoding aqubit is present on each input path of the multiplexer circuit; select,based on the input signal, one of the input paths of the multiplexercircuit as an active input path for the time bin; select one of theoutput paths of the multiplexer circuit as an active output path for thetime bin, wherein output paths in the first and second subsets of outputpaths are selected according to a fixed order such that each output pathin the first subset of output paths and each output path in the secondsubset of output paths is selected as the active output path once duringa raster period consisting of 2R consecutive time bins; and generatecontrol signals to set a state of the optical switching network of themultiplexer circuit for the time bin such that a photon from the activeinput path is coupled to the active output path.
 4. The system of claim3 wherein the control logic is further configured to: select, for eachtime bin, an active output path for each of the first optical switchingcircuit and the second optical switching circuit such that a qubitoutput on the active output path of the first optical switching circuitand a qubit output on the active output path of the second opticalswitching circuit arrive in temporal alignment at one of the downstreamcircuits.
 5. The system of claim 3 wherein the plurality of output pathsof the first optical switching circuit and the plurality of output pathsof the second optical switching circuit each include 2R output paths. 6.The system of claim 5 wherein the delay circuits introduce differentdelays in a range from 0 to 2(R−1) time bins.
 7. The system of claim 2wherein the plurality of downstream circuits includes a plurality offusion circuits configured to perform a joint measurement operation thatconsumes a pair of qubits received at the first and second input portsand produces measurement data.
 8. The system of claim 7 wherein thejoint measurement operation is a Type II fusion operation.
 9. The systemof claim 7 wherein the plurality of fusion circuits includes a number offusion circuits equal to 2R−1.
 10. The system of claim 2 furthercomprising: a first single-qubit measurement circuit; and a secondsingle-qubit measurement circuit, wherein the first optical switchingcircuit includes an output path coupled to an input path of the firstsingle-qubit measurement circuit and the second optical switchingcircuit includes an output path coupled to an input path of the secondsingle-qubit measurement circuit.
 11. The system of claim 2 wherein eachinput path and each output path of the multiplexer circuit and the firstand second optical switching circuit comprises a waveguide.
 12. Thesystem of claim 2 wherein each input path and each output path of themultiplexer circuit and the first and second optical switching circuitcomprises a pair of waveguides.
 13. The system of claim 2 wherein theoptical switching network of the multiplexer circuit is a generalizedMach-Zehnder interferometer (GMZI) and the active optical switchesinclude active phase shifters.
 14. The system of claim 2 wherein each ofthe first and second optical switching circuits includes a generalizedMach-Zehnder interferometer (GMZI).
 15. A method comprising: receiving,at a first optical switching circuit, a first group of qubits via afirst plurality of optical paths, wherein during each of a plurality ofsuccessive time bins in a first set of time bins, each optical pathreceives one qubit of the first group of qubits; receiving, at a secondoptical switching circuit, a second group of qubits via a secondplurality of optical paths, wherein during each of a plurality ofsuccessive time bins in a second set of time bins, each optical pathreceives one qubit of the second group of qubits; identifying a targetpairing between a first qubit in the first group of qubits and a secondqubit in the second group of qubits; routing, by the first opticalswitching circuit, the first qubit to a selected one of a plurality ofdownstream circuits via one of a plurality of output paths of the firstoptical switching circuit, wherein different ones of the output paths ofthe first optical switching circuit incorporate different amounts ofdelay; and routing, by the second optical switching circuit, the secondqubit to the selected one of the plurality of downstream circuits viaone of a plurality of output paths of the second optical switchingcircuit such that the first qubit and the second qubit arrive at theselected one of the plurality of downstream circuits in temporalalignment.
 16. The method of claim 15 further comprising: providingphotonic states encoding qubits from a plurality of source circuits on aplurality of input paths to a multiplexer circuit having a plurality ofoutput paths that includes a first subset of output paths coupled to thefirst plurality of optical paths and a second subset of output pathscoupled to the second plurality of optical paths; and operating themultiplexer circuit for the first set of time bins and the second set oftime bins such that, for each time bin, a qubit from one of the sourcecircuits is delivered to a different one of the plurality of outputpaths of the multiplexer circuit.
 17. The method of claim 16 wherein thefirst subset of output paths of the multiplexer circuit and the secondsubset of output paths of the multiplexer circuit each include a number(R) of output paths and wherein operating the multiplexer circuitincludes: receiving, for each time bin, an input signal indicative ofwhether a photonic state encoding a qubit is present on each input pathof the multiplexer circuit; selecting, based on the input signal, one ofthe input paths of the multiplexer circuit as an active input path forthe time bin; selecting one of the output paths of the multiplexercircuit as an active output path for the time bin, wherein output pathsin the first and second subsets of output paths are selected accordingto a fixed order such that each output path in the first subset ofoutput paths and each output path in the second subset of output pathsis selected as the active output path once during a raster periodconsisting of 2R consecutive time bins; and generating control signalsto set a state of an optical switching network of the multiplexercircuit for the time bin such that a photon from the active input pathis coupled to the active output path.
 18. The method of claim 17 whereinthe different ones of the output paths of the first optical switchingcircuit incorporate different amounts of delay in a range from 0 to2(R−1) time bins.
 19. The method of claim 15 further comprising:operating the downstream circuits to perform joint measurementoperations on pairs of qubits received in temporal alignment.
 20. Themethod of claim 19 wherein the joint measurement operation is a Type IIfusion operation.
 21. The method of claim 15 further comprising:operating one or more of the downstream circuits to perform asingle-qubit measurement on a received qubit.